#6584745

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1863 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)(13,14,19,23)(15,20,25,34)(16,27,30,41)(17,22,18,32)(21,28,37,29)(24,38,39,44)(26,35,31,36)(33,42,43,40);(1,2)(3,4)(6,10)(11,12)(13,15,19,25)(14,20,23,34)(16,24,30,39)(17,31,18,26)(21,33,37,43)(22,36,32,35)(27,38,41,44)(28,42,29,40);(2,4)(5,7,9,8)(6,11,10,12)(13,16,18,29,19,30,17,28)(14,21,22,27,23,37,32,41)(15,24,26,40,25,39,31,42)(20,33,36,38,34,43,35,44);(1,2)(3,4)(5,8,9,7)(6,12,10,11)(15,25)(20,34)(24,39)(26,31)(33,43)(35,36)(38,44)(40,42);(2,4)(13,17,19,18)(14,22,23,32)(15,26,25,31)(16,28,30,29)(20,35,34,36)(21,27,37,41)(24,40,39,42)(33,44,43,38)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,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=ehi,d^c=dh,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, 6491941);