#354510

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

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 ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1004 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)(5,6)(7,11)(8,12)(9,10)(13,14)(15,20)(16,28)(17,31)(18,23)(19,22)(21,29)(24,38)(25,41)(26,35)(27,34)(30,37)(32,36)(33,39)(40,44)(42,43);(6,10)(11,12)(13,15,18,26,19,27,17,25)(14,20,23,35,22,34,31,41)(16,24,29,39,30,40,32,42)(21,33,37,44,36,43,28,38);(5,7,9,8)(6,11,10,12)(13,16)(14,21)(15,24)(17,32)(18,29)(19,30)(20,33)(22,36)(23,37)(25,42)(26,39)(27,40)(28,31)(34,43)(35,44)(38,41);(1,3,4,2)(5,8,9,7)(6,12,10,11)(14,22)(20,34)(21,36)(23,31)(28,37)(33,43)(35,41)(38,44)

Polycyclic presentation:

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

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, 261706);