#474120

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
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 ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

944 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)(5,6,8,10)(7,9,11,12)(14,17)(18,20)(22,26)(23,29)(25,31)(27,34)(30,36)(32,35);(1,2,4,3)(5,7,8,11)(6,9,10,12)(13,14)(15,18)(16,17)(19,20)(21,22,24,26)(23,27,30,32)(25,28,31,33)(29,35,36,34);(2,3)(7,11)(9,12)(14,17)(15,19)(21,23,25,29,24,30,31,36)(22,27,28,35,26,32,33,34);(1,3,4,2)(13,15,16,19)(14,18,17,20)(22,26)(23,30)(28,33)(29,36)

Polycyclic presentation:

2^9|a^2=egi,b^2=egh,c^2=f,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,d^c=de

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 381316);