#6611806

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	#2615	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1859 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)(12,16)(13,15)(17,18,22,26)(19,27,30,40)(20,31,32,24)(21,34,36,25)(23,35,38,29)(28,42,43,41)(33,39,47,46)(37,44,45,48);(9,11,13,12)(10,14,15,16)(17,19,22,30)(18,23,26,38)(20,28,32,43)(21,35,36,29)(24,37,31,45)(25,40,34,27)(33,48,47,44)(39,42,46,41);(2,5)(6,8)(10,15)(11,12)(17,20)(18,24)(19,28)(21,33)(22,32)(23,37)(25,39)(26,31)(27,41)(29,44)(30,43)(34,46)(35,48)(36,47)(38,45)(40,42);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,15,14)(20,32)(24,31)(28,43)(33,47)(37,45)(39,46)(41,42)(44,48);(11,12)(14,16)(20,32)(24,31)(28,43)(33,47)(37,45)(39,46)(41,42)(44,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6519002);