#6552686

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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	#38035	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1832 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fi,e^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^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6459882);