#6558244

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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

2560 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)(25,39)(26,35)(28,34)(29,36)(30,37)(31,38)(33,40)(41,43)(42,44);(5,6)(7,10)(8,9)(11,12)(13,15,18,27,19,28,17,26)(14,20,22,24,23,35,32,34)(16,25,29,40,31,41,30,42)(21,33,37,39,38,44,36,43);(1,2,3,4)(6,9)(10,12)(13,16)(14,21)(15,25)(17,30)(18,29)(19,31)(20,33)(22,37)(23,38)(24,39)(26,42)(27,40)(28,41)(32,36)(34,43)(35,44);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,32)(15,26,28,27)(16,29,31,30)(20,24,35,34)(21,36,38,37)(25,40,41,42)(33,43,44,39);(7,11)(10,12)(13,17,19,18)(14,22,23,32)(15,27,28,26)(16,30,31,29)(20,34,35,24)(21,37,38,36)(25,40,41,42)(33,43,44,39)

Polycyclic presentation:

2^9|b^2=f,c^2=g,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,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, 6465440);