#6579341

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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	#39356	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1994 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)(13,14,18,23)(15,24,28,38)(16,30,32,36)(17,21,25,22)(19,27,34,29)(20,35,37,31)(26,40,42,33)(39,41,44,43);(1,2)(3,4)(6,9)(10,12)(13,15,25,27,18,28,17,29)(14,19,21,38,23,34,22,24)(16,26,35,41,32,42,31,43)(20,33,36,44,37,40,30,39);(2,4)(5,7,8,11)(6,10,9,12)(13,16,17,31,18,32,25,35)(14,20,22,30,23,37,21,36)(15,26,29,43,28,42,27,41)(19,33,24,39,34,40,38,44);(1,2)(3,4)(13,17,18,25)(14,21,23,22)(15,27,28,29)(16,31,32,35)(19,24,34,38)(20,36,37,30)(26,41,42,43)(33,39,40,44);(2,4)(15,28)(19,34)(24,38)(26,42)(27,29)(33,40)(39,44)(41,43)

Polycyclic presentation:

2^9|a^2=i,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=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, 6486537);