#6456416

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1864 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=fi,e^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6363612);