#96797

order	512
nilpotency class	3
isoclinism	isoclinic class of #7654
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₄³	C₂³
socle	C₂³	C₄×C₂²⋊C₄
prime radical	C₂²×D₄	C₄²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

397 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 397 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fgh,c^b=cghi,d^b=dh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3993);