#6434708

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432876
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2852 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)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18,23,28)(19,29,32,41)(20,25,34,40)(21,27,22,26)(24,35,38,31)(30,44,47,43)(33,42,36,39)(37,45,48,46);(1,3,5,10)(2,6,8,14)(4,11,12,9)(7,15,16,13)(17,19)(18,24)(20,30)(21,31)(22,35)(23,32)(25,37)(26,29)(27,41)(28,38)(33,45)(34,47)(36,46)(39,43)(40,48)(42,44);(1,4,5,12)(2,7,8,16)(3,9,10,11)(6,13,14,15)(17,20)(18,25)(19,30)(21,33)(22,36)(23,34)(24,37)(26,39)(27,42)(28,40)(29,43)(31,45)(32,47)(35,46)(38,48)(41,44);(2,8)(6,14)(7,16)(13,15);(17,21,23,22)(18,26,28,27)(19,31,32,35)(20,33,34,36)(24,29,38,41)(25,39,40,42)(30,45,47,46)(37,43,48,44)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,f^a=fi,c^b=ch,f^b=fi,g^b=gi,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, 6341904);