#125075

order	512
nilpotency class	4
isoclinism	isoclinic class of #125068
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#95	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂×#95 ▷ C₂⁴ ▷ C₂ ▷ C₁

454 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,5)(4,7)(6,8)(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24)(25,26,29,33,30,32,37,39)(27,34,35,40,28,31,36,38);(2,5)(3,8)(6,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,28)(26,31,32,34)(29,35,37,36)(33,40,39,38);(11,17)(14,20)(18,23)(21,24)(25,28,30,27)(26,31,32,34)(29,36,37,35)(33,38,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32271);