#122780

order	512
nilpotency class	4
isoclinism	isoclinic class of #122694
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
prime radical	#24265	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 458 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29976);