#512879

order	512
nilpotency class	5
isoclinism	isoclinic class of #512876
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
prime radical	#61100	C₂
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

1014 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 1014 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,g^a=gh,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420075);