#96833

order	512
nilpotency class	3
isoclinism	isoclinic class of #1510
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	C₄×C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

368 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eh,d^2=g,e^2=i,f^2=h,b^a=bd,d^a=dg,f^a=fgh,c^b=cg,d^b=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4029);