#355020

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×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

1010 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ef,d^2=eh,f^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,d^c=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, 262216);