#149662

order	512
nilpotency class	4
isoclinism	isoclinic class of #11521
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#410
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×M₃₂	C₂³
socle	C₂	#11460
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₄×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

205 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 205 conjugacy classes of subgroups — too many to draw. Only its 72 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=f,d^2=i,e^2=g,g^2=h,h^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fi,c^b=ci,d^b=di,e^b=efi,e^c=ei,e^d=ei

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, 56858);