#124901

order	512
nilpotency class	4
isoclinism	isoclinic class of #124835
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#329
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

301 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 301 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,4)(7,8)(9,10,12,18,14,17,29,34,15,20,27,37,13,19,28,36)(11,21,22,38,24,33,26,40,25,32,39,35,23,16,30,31);(1,2)(3,6)(4,8)(5,7)(9,11,15,25)(10,16,20,33)(12,26,27,30)(13,24,14,23)(17,32,19,21)(18,35,37,38)(22,28,39,29)(31,34,40,36);(1,3,5,4)(2,6,7,8)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fh,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cfh,d^a=df,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,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, 32097);