#125808

order	512
nilpotency class	4
isoclinism	isoclinic class of #125725
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂×#97	C₂³
prime radical	#3150	C₂²
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

331 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 331 conjugacy classes of subgroups — too many to draw. Only its 56 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=gi,d^2=f,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,g^a=gi,h^a=hi,d^b=df,e^b=egi,d^c=dg,e^c=ehi,h^c=hi,e^d=ei

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33004);