#6508687

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499844
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62821	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2972 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 6415883);