#125290

order	512
nilpotency class	4
isoclinism	isoclinic class of #125287
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

377 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 377 conjugacy classes of subgroups — too many to draw. Only its 82 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)(9,10,13,19,15,22,32,18,16,20,31,38,14,21,12,30)(11,23,26,35,28,39,25,36,29,33,40,17,27,34,24,37);(1,2)(3,6)(4,8)(5,7)(9,11,16,29)(10,17,20,35)(12,26,32,40)(13,24,31,25)(14,28,15,27)(18,34,30,39)(19,23,38,33)(21,37,22,36);(1,3,5,4)(2,6,7,8)(9,12,15,13,16,32,14,31)(10,18,22,38,20,30,21,19)(11,24,28,26,29,25,27,40)(17,33,37,34,35,23,36,39)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32486);