#512883

order	512
nilpotency class	5
isoclinism	isoclinic class of #512880
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	#124	C₂⁴
LCS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #532 ◁ G
LECS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1022 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 1022 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)(3,14)(4,10)(5,11)(6,25)(7,28)(8,13)(9,17)(12,30)(15,44)(16,33)(18,43)(19,38)(20,34)(21,54)(22,36)(23,37)(24,41)(26,57)(27,39)(29,40)(31,46)(32,52)(35,56)(42,47)(45,60)(48,63)(49,64)(50,58)(51,59)(53,62)(55,61);(1,3,8,19)(2,9,13,33)(4,15,23,49)(5,17,6,16)(7,18,30,53)(10,31,37,61)(11,38,25,14)(12,32,28,58)(20,51,26,45)(21,55,56,46)(22,48,29,47)(24,52,27,50)(34,63,57,42)(35,64,54,44)(36,60,40,59)(39,62,41,43);(1,4)(2,10)(3,15)(5,20)(6,26)(7,29)(8,23)(9,31)(11,34)(12,40)(13,37)(14,42)(16,45)(17,51)(18,47)(19,49)(21,27)(22,30)(24,56)(25,57)(28,36)(32,59)(33,61)(35,39)(38,63)(41,54)(43,44)(46,52)(48,53)(50,55)(58,60)(62,64);(1,5,8,6)(2,11,13,25)(3,16,19,17)(4,20,23,26)(7,24,30,27)(9,14,33,38)(10,34,37,57)(12,39,28,41)(15,45,49,51)(18,50,53,52)(21,29,56,22)(31,42,61,63)(32,43,58,62)(35,36,54,40)(44,60,64,59)(46,47,55,48)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209895
maximal normal subgroup of	#72209895
maximal quotient of	#72227843, #72228006, #72228033

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420079);