#6455583

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

1507 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)(19,28)(20,26)(21,36)(22,25)(23,27)(24,32)(29,38)(30,45)(31,40)(33,48)(34,41)(35,44)(37,43)(39,47)(42,46);(2,5)(6,8)(17,19)(18,24)(20,33)(21,31)(22,32)(23,29)(25,28)(26,42)(27,40)(30,35)(34,46)(36,38)(37,47)(39,44)(41,48)(43,45);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(18,25)(19,29)(20,34)(21,23)(24,38)(28,40)(30,46)(31,32)(33,47)(39,42)(43,44)(45,48);(17,20,22,34)(18,26,25,41)(19,30,32,47)(21,35,23,37)(24,39,28,45)(27,43,36,44)(29,33,31,46)(38,48,40,42);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,27,25,36)(19,31,32,29)(20,35,34,37)(24,40,28,38)(26,43,41,44)(30,46,47,33)(39,42,45,48)

Polycyclic presentation:

2^9|c^2=fi,d^2=i,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,e^c=ei,h^c=hi,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, 6362779);