#125158

order	512
nilpotency class	4
isoclinism	isoclinic class of #125092
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1530
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	#33179	C₂
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

433 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 433 conjugacy classes of subgroups — too many to draw. Only its 92 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)(9,10,12,19,14,18,25,33,15,20,26,34,13,16,27,29)(11,17,22,31,21,30,35,39,24,32,36,40,23,28,37,38);(1,3)(2,5)(4,8)(6,7)(10,16)(11,21)(12,25)(13,14)(18,20)(19,34)(22,36)(23,24)(26,27)(28,30)(31,38)(39,40);(9,11,13,23,15,24,14,21)(10,17,16,28,20,32,18,30)(12,22,27,37,26,36,25,35)(19,31,29,38,34,40,33,39)

Polycyclic presentation:

2^9|a^2=e,c^2=hi,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,d^a=df,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,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, 32354);