#6357711

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,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, 6264907);