#124645

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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,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, 31841);