#408240

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1354 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=efhi,d^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 315436);