#357269

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

942 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,4,3)(6,10)(11,12)(13,14,18,22,19,23,17,31)(15,24,26,20,27,34,32,35)(16,21,30,37,28,36,29,41)(25,38,40,33,39,44,42,43);(1,2,4,3)(5,6)(7,11)(8,12)(9,10)(16,28)(21,36)(25,39)(29,30)(33,43)(37,41)(38,44)(40,42);(1,3,4,2)(5,7,9,8)(6,11,10,12)(13,15,18,26,19,27,17,32)(14,20,22,34,23,35,31,24)(16,25,30,40,28,39,29,42)(21,33,37,44,36,43,41,38);(5,8,9,7)(6,11,10,12)(13,16,18,30,19,28,17,29)(14,21,22,37,23,36,31,41)(15,25,26,40,27,39,32,42)(20,33,34,44,35,43,24,38)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=eg,d^2=egh,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,d^b=dei,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264465);