#149660

order	512
nilpotency class	4
isoclinism	isoclinic class of #11522
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	C₂⁴	M₃₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₄×C₈ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×M₃₂ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 203 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56856);