#324792

order	512
nilpotency class	3
isoclinism	isoclinic class of #278011
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

617 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 231988);