#6558172

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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

2509 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)(13,14,19,23)(15,24,28,41)(16,21,31,38)(17,22,18,33)(20,26,35,27)(25,40,29,44)(30,37,32,39)(34,43,36,42);(5,6)(7,10)(8,9)(11,12)(13,15,17,26,19,28,18,27)(14,20,33,41,23,35,22,24)(16,29,32,42,31,25,30,43)(21,36,37,40,38,34,39,44);(1,2,3,4)(6,9)(10,12)(13,16)(14,21)(15,25)(17,32)(18,30)(19,31)(20,34)(22,39)(23,38)(24,40)(26,43)(27,42)(28,29)(33,37)(35,36)(41,44);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,33)(15,26,28,27)(16,30,31,32)(20,24,35,41)(21,37,38,39)(25,42,29,43)(34,44,36,40);(7,11)(10,12)(16,31)(21,38)(25,29)(30,32)(34,36)(37,39)(40,44)(42,43)

Polycyclic presentation:

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

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, 6465368);