#321384

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=egi,c^2=i,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 228580);