#416762

order	512
nilpotency class	3
isoclinism	isoclinic class of #393940
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

982 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)(7,11,12,9)(13,14,16,17)(15,19,20,18)(21,22)(23,26)(24,28)(25,27)(29,33)(30,35)(31,34)(32,36);(1,2,4,3)(5,7,8,12)(6,9,10,11)(14,17)(18,19)(21,23,25,31)(22,26,27,34)(24,29,32,30)(28,33,36,35);(1,3,4,2)(23,29)(24,32)(26,33)(28,36)(30,31)(34,35);(13,15,16,20)(14,18,17,19)(22,27)(26,34)(28,36)(33,35)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323958);