#6351304

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351101
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2591 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258500);