#124923

order	512
nilpotency class	4
isoclinism	isoclinic class of #124851
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1505
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

277 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=fghi,d^2=g,e^2=h,g^2=i,h^2=i,b^a=bd,c^a=cfh,d^a=df,f^a=fi,d^b=dg,e^b=efgi,g^b=gi,h^b=hi,d^c=di,e^d=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, 32119);