#6350951

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

2640 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)(3,9,13,15)(4,14,12,8)(5,10,6,16)(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);(1,3,7,13)(2,8,11,14)(4,5,12,6)(9,10,15,16)(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)(2,9)(3,5)(6,13)(7,12)(8,10)(11,15)(14,16)(17,20)(18,25)(19,29)(21,33)(22,34)(23,32)(24,37)(26,41)(27,40)(28,42)(30,44)(31,43)(35,45)(36,46)(38,48)(39,47);(1,2)(3,8)(4,15)(5,16)(6,10)(7,11)(9,12)(13,14)(18,26)(20,32)(24,38)(27,35)(28,39)(29,43)(33,34)(44,46);(1,5,7,6)(2,10,11,16)(3,12,13,4)(8,15,14,9)(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,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, 6258147);