#6358050

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358021
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

1874 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=gh,d^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=dhi,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, 6265246);