#6357709

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357668
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

1836 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264905);