#400359

order	512
nilpotency class	3
isoclinism	isoclinic class of #393944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

974 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:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=ei,d^2=efi,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, 307555);