#124637

order	512
nilpotency class	3
isoclinism	isoclinic class of #7694
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×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

294 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,4,6,5,7,3,8)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22)(25,26,29,35,31,32,38,39,30,34,37,40,27,36,28,33);(2,6)(3,4)(7,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27,30,31)(26,32,34,36)(28,29,38,37)(33,40,39,35);(2,7)(6,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,27,37,30,38,31,29)(26,33,36,40,34,39,32,35)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=fgh,d^2=g,e^2=h,h^2=i,b^a=bd,c^a=cg,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, 31833);