#512906

order	512
nilpotency class	5
isoclinism	isoclinic class of #512902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2196
commutator, Frattini, cosocle	C₄⋊Q₈	C₂⁴
prime radical	#61111	C₂
LCS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1009 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 1009 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,9,16)(3,17,22,44)(4,23,27,45)(5,15,8,12)(6,14,7,13)(10,31,37,20)(11,32,40,26)(18,49,54,63)(19,36,28,48)(21,35,55,41)(24,58,33,39)(25,46,59,38)(29,42,30,43)(34,53,64,50)(47,51,57,52)(56,62,60,61);(1,3,9,22)(2,10,16,37)(4,18,27,54)(5,28,8,19)(6,31,7,20)(11,34,40,64)(12,41,15,35)(13,44,14,17)(21,29,55,30)(23,57,45,47)(24,53,33,50)(25,52,59,51)(26,60,32,56)(36,42,48,43)(38,63,46,49)(39,62,58,61);(1,4)(2,11)(3,18)(5,24)(6,26)(7,32)(8,33)(9,27)(10,34)(12,38)(13,23)(14,45)(15,46)(16,40)(17,47)(19,50)(20,56)(21,51)(22,54)(25,29)(28,53)(30,59)(31,60)(35,49)(36,61)(37,64)(39,42)(41,63)(43,58)(44,57)(48,62)(52,55);(1,5,9,8)(2,12,16,15)(3,19,22,28)(4,24,27,33)(6,29,7,30)(10,35,37,41)(11,38,40,46)(13,42,14,43)(17,48,44,36)(18,50,54,53)(20,55,31,21)(23,39,45,58)(25,32,59,26)(34,49,64,63)(47,62,57,61)(51,56,52,60)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,g^a=gh,d^b=di,e^b=ei,f^b=fghi,h^b=hi,e^c=eg,f^c=fi,g^c=gi,h^c=hi,f^e=fhi,g^e=gi,g^f=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209895
maximal normal subgroup of	#72209895
maximal quotient of	#72227838, #72228005, #72228033

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420102);