#125042

order	512
nilpotency class	4
isoclinism	isoclinic class of #124934
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#331
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

356 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 90 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fghi,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cfgh,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi

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, 32238);