#285777

order	512
nilpotency class	3
isoclinism	isoclinic class of #281659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

674 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=eh,d^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192973);