#124522

order	512
nilpotency class	4
isoclinism	isoclinic class of #11992
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1555
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

474 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31718);