#6352019

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351885
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

2645 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)(3,13)(4,14)(5,10)(6,11)(7,12)(8,15)(9,16)(18,23)(20,30)(21,32)(24,37)(25,40)(26,33)(27,42)(28,34)(35,39)(36,41)(38,46)(47,48);(1,3,7,9)(2,8,12,4)(5,14,6,15)(10,13,11,16)(17,18,22,26)(19,24,29,39)(20,25,30,41)(21,33,32,23)(27,38,42,48)(28,35,34,37)(31,40,44,36)(43,46,45,47);(1,4,7,8)(2,9,12,3)(5,16,6,13)(10,15,11,14)(17,19,22,29)(18,24,26,39)(20,27,30,42)(21,34,32,28)(23,35,33,37)(25,38,41,48)(31,45,44,43)(36,46,40,47);(1,2)(3,8)(4,9)(5,11)(6,10)(7,12)(13,15)(14,16)(17,20)(18,25)(19,27)(21,31)(22,30)(23,36)(24,38)(26,41)(28,43)(29,42)(32,44)(33,40)(34,45)(35,46)(37,47)(39,48);(1,5,7,6)(2,10,12,11)(3,14,9,15)(4,16,8,13)(20,30)(25,41)(27,42)(31,44)(36,40)(38,48)(43,45)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259215);