#6558124

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2504 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6465320);