#354156

order	512
nilpotency class	3
isoclinism	isoclinic class of #3003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄³

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	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1304 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,f^c=fi,g^c=gi

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, 261352);