#124376

order	512
nilpotency class	3
isoclinism	isoclinic class of #7694
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 272 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) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=fg,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=cfi,d^a=df,c^b=ci,d^b=dg,e^b=efg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31572);