#94810

order	512
nilpotency class	6
isoclinism	isoclinic class of #94810
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₄×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7504
commutator	C₄×C₈	C₄²
Frattini, prime radical, cosocle	#3147	C₂²
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #3147 ◁ G
LECS	G ▷ #3147 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

190 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 36 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 — 2 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=ghi,e^2=h,f^2=hi,g^2=i,b^a=bc,c^a=cf,e^a=egi,f^a=fg,g^a=gh,h^a=hi,c^b=ch,d^b=dfgi,f^b=fi,d^c=dgh,e^d=ehi,f^d=fhi,g^d=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2006);