#6352848

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352701
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	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2715 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6260044);