#125318

order	512
nilpotency class	4
isoclinism	isoclinic class of #125297
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

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

267 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=fgi,d^2=gi,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cf,d^a=df,g^a=gi,c^b=ch,d^b=dg,e^b=efg,f^b=fi,h^b=hi,d^c=di,e^c=ei,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, 32514);