#6350743

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350717
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

2689 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,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

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257939);