#6584773

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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	#37701	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1895 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491969);