#356587

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle, prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1126 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ef,d^2=fg,g^2=i,b^a=be,c^a=cf,d^b=dei

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263783);