#357275

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₄³

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

1166 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=eg,d^2=efgh,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, 264471);