#6353282

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

2744 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)(4,5)(8,15)(9,12)(10,13)(14,16)(17,18,22,26)(19,27,30,40)(20,31,32,24)(21,34,36,25)(23,35,38,29)(28,42,43,41)(33,39,47,46)(37,44,45,48);(1,2,6,9)(3,8,11,16)(4,7,5,12)(10,14,13,15)(17,19,22,30)(18,23,26,38)(20,28,32,43)(21,35,36,29)(24,37,31,45)(25,40,34,27)(33,48,47,44)(39,42,46,41);(1,3,6,11)(2,8,9,16)(4,10,5,13)(7,14,12,15);(1,4,6,5)(2,7,9,12)(3,10,11,13)(8,14,16,15)(17,20,22,32)(18,24,26,31)(19,28,30,43)(21,33,36,47)(23,37,38,45)(25,39,34,46)(27,41,40,42)(29,44,35,48);(20,32)(24,31)(28,43)(33,47)(37,45)(39,46)(41,42)(44,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^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, 6260478);