#355644

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

1002 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)(8,10)(9,12)(13,14)(15,18)(16,17)(19,20)(21,22,25,24,26,29,27,30)(23,31,35,36,34,28,33,32);(5,7)(6,9)(8,12)(10,11)(21,23,27,33,26,34,25,35)(22,28,30,36,29,31,24,32);(1,3,4,2)(7,12)(9,11)(14,17)(18,20)(21,24,25,29,26,30,27,22)(23,32,35,31,34,36,33,28);(6,10)(9,11)(13,15,16,19)(14,18,17,20)(21,25,26,27)(22,24,29,30)(23,33,34,35)(28,36,31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cei,d^b=di

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, 262840);