#125404

order	512
nilpotency class	4
isoclinism	isoclinic class of #125393
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

444 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fi,c^2=i,d^2=gi,e^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fi,c^b=ch,d^b=dg,e^b=efgi,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, 32600);