#414463

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
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

1048 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)(5,6,8,10)(7,11,12,9)(23,29)(24,32)(26,33)(28,34)(30,35)(31,36);(5,7,8,12)(6,9,10,11)(13,14)(15,18)(16,17)(19,20)(21,22,25,27)(23,30,31,26)(24,28,32,34)(29,35,36,33);(14,17)(18,20)(21,23,24,29,25,31,32,36)(22,26,28,33,27,30,34,35);(1,2,3,4)(13,15,16,19)(14,18,17,20)(22,27)(26,30)(28,34)(33,35)

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=f,d^2=ghi,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=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, 321659);