#6579377

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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	C₂×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1992 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)(7,10)(8,9)(11,12)(13,14,18,22)(15,23,26,40)(16,28,30,43)(17,32,24,21)(19,33,35,27)(20,36,38,31)(25,37,29,34)(39,44,42,41);(1,2)(3,4)(6,9)(10,12)(13,15,24,33,18,26,17,27)(14,19,32,40,22,35,21,23)(16,29,36,41,30,25,31,44)(20,37,43,39,38,34,28,42);(2,4)(5,7,8,11)(6,10,9,12)(13,16,17,31,18,30,24,36)(14,20,21,28,22,38,32,43)(15,25,27,41,26,29,33,44)(19,34,23,39,35,37,40,42);(1,2)(3,4)(16,30)(20,38)(25,29)(28,43)(31,36)(34,37)(39,42)(41,44);(2,4)(15,26)(19,35)(23,40)(25,29)(27,33)(34,37)(39,42)(41,44)

Polycyclic presentation:

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

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, 6486573);