#404165

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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 ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1364 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)(5,6,8,10)(7,11,12,9)(21,22,24,28)(23,30,31,26)(25,27,33,29)(32,35,36,34);(5,7,8,12)(6,9,10,11)(13,14)(15,18)(16,17)(19,20)(21,23)(22,26)(24,31)(25,32)(27,34)(28,30)(29,35)(33,36);(14,17)(18,20)(22,27)(25,33)(26,34)(28,29)(30,35)(32,36);(2,4)(13,15,16,19)(14,18,17,20)(23,31)(26,30)(32,36)(34,35)

Polycyclic presentation:

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

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, 311361);