#6358017

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357957
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1815 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:

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

Polycyclic presentation:

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

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, 6265213);