#6350955

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350909
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

2664 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,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, 6258151);