#478240

order	512
nilpotency class	3
isoclinism	isoclinic class of #476082
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

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

1082 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)(6,9)(10,11)(13,14,16,17)(15,19,20,18)(23,29)(26,35)(30,36)(31,34);(5,6,8,9)(7,11,12,10)(14,17)(18,19)(21,22,24,27)(23,30,29,36)(25,33,32,28)(26,34,35,31);(7,12)(10,11)(13,15,16,20)(14,18,17,19)(21,23,24,29)(22,26,27,35)(25,34,32,31)(28,36,33,30);(2,4)(5,7,8,12)(6,10,9,11)(15,20)(18,19)(22,27)(26,35)(28,33)(30,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,c^2=f,d^2=efi,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi

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, 385436);