#122758

order	512
nilpotency class	4
isoclinism	isoclinic class of #122714
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄⋊C₈	C₂³
prime radical	C₂²×D₄	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

434 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 434 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=gh,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,c^b=cfi,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,d^c=dhi,e^d=ehi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29954);