#6456085

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1466 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,27)(19,24,31,40)(20,32,34,25)(21,36,23,26)(28,45,47,38)(29,39,30,41)(33,48,46,42)(35,43,37,44);(2,5)(6,8)(17,19,22,31)(18,24,27,40)(20,33,34,46)(21,30,23,29)(25,42,32,48)(26,39,36,41)(28,37,47,35)(38,43,45,44);(10,13)(14,16)(17,20,22,34)(18,25,27,32)(19,28,31,47)(21,35,23,37)(24,38,40,45)(26,43,36,44)(29,33,30,46)(39,42,41,48);(1,3,4,7)(2,6,5,8)(19,29)(20,34)(21,23)(24,39)(25,32)(26,36)(28,46)(30,31)(33,47)(38,48)(40,41)(42,45);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,26,27,36)(19,30,31,29)(20,35,34,37)(24,39,40,41)(25,43,32,44)(28,46,47,33)(38,42,45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,d^c=di,h^c=hi,e^d=ei,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6363281);