#357385

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂×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

994 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)(5,6)(7,11)(8,12)(9,10)(13,14)(15,17)(16,18)(19,20)(21,22,26,31,25,30,27,32)(23,28,24,35,33,36,34,29);(6,10)(7,8)(13,15)(14,17)(16,19)(18,20)(21,23,27,34,25,33,26,24)(22,28,32,29,30,36,31,35);(1,3,4,2)(5,7,9,8)(6,11,10,12)(14,18)(15,19)(21,24,26,33,25,34,27,23)(22,29,31,28,30,35,32,36);(5,8,9,7)(6,11,10,12)(14,18)(17,20)(22,30)(28,36)(29,35)(31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=eg,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cefi,d^b=de

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264581);