#6558188

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535655
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2561 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=g,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=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, 6465384);