#6579387

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#39357	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6486583);