#122485

order	512
nilpotency class	4
isoclinism	isoclinic class of #11677
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₈²	C₂³
socle	C₂²	C₂×D₄⋊C₈
prime radical	C₂³	D₄⋊C₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

242 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 256

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=h,d^2=f,e^2=g,f^2=h,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,c^b=chi,d^b=dfh,e^b=eh,f^b=fh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29681);