#125730

order	512
nilpotency class	4
isoclinism	isoclinic class of #125729
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7737
commutator, Frattini, cosocle	C₂×#97	C₂³
LCS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

383 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219621, #72226635, #72226637
maximal normal subgroup of	#72219621, #72226635, #72226637

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32926);