#406705

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, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1104 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:

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

Polycyclic presentation:

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

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, 313901);