#6516602

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499814
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1354 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=fhi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,g^c=gi

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, 6423798);