#6353219

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
prime radical	#62960	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2772 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)(3,11)(4,9)(5,15)(6,10)(7,12)(8,13)(14,16)(17,18,23,22)(19,24,30,29)(20,25,33,32)(21,26,34,35)(27,39,41,36)(28,37,42,43)(31,38,44,45)(40,47,48,46);(1,3)(2,7)(4,12)(5,14)(6,13)(8,9)(10,16)(11,15)(17,19)(18,24)(20,27)(21,28)(22,29)(23,30)(25,36)(26,37)(31,40)(32,39)(33,41)(34,42)(35,43)(38,46)(44,48)(45,47);(17,20,23,33)(18,25,22,32)(19,27,30,41)(21,31,34,44)(24,36,29,39)(26,38,35,45)(28,40,42,48)(37,46,43,47);(1,3)(2,8)(4,14)(5,12)(6,13)(7,9)(10,11)(15,16)(17,21,23,34)(18,26,22,35)(19,28,30,42)(20,31,33,44)(24,37,29,43)(25,38,32,45)(27,40,41,48)(36,46,39,47);(1,4,6,5)(2,9,10,15)(3,12,13,14)(7,8,16,11)(17,22,23,18)(19,29,30,24)(20,32,33,25)(21,26,34,35)(27,39,41,36)(28,37,42,43)(31,38,44,45)(40,46,48,47)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=i,e^2=hi,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, 6260415);