#122533

order	512
nilpotency class	4
isoclinism	isoclinic class of #11858
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 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

236 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 86 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)(33,34,36,41,38,42,49,56,37,40,48,53,50,54,61,63)(35,43,45,55,46,57,59,64,44,39,47,51,58,52,60,62);(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)(36,47)(37,44)(38,46)(40,43)(41,55)(42,52)(45,48)(49,60)(50,58)(51,53)(54,57)(56,64)(59,61)(62,63);(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)(35,44)(41,53)(42,54)(45,47)(46,58)(56,63)(59,60)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=h,d^2=fi,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=dfhi,e^b=ehi,f^b=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29729);