#6470586

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470149
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	#37445	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1777 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 6377782);