#6353234

order	512
nilpotency class	3
isoclinism	isoclinic class of #6353061
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×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2718 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:

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

Polycyclic presentation:

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

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, 6260430);