#6353109

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×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2673 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)(19,28)(20,32)(21,26)(22,36)(23,27)(24,30)(25,34)(29,44)(31,39)(33,42)(35,41)(37,40)(38,45)(43,47)(46,48);(1,2,6,9)(3,8,11,16)(4,7,5,12)(10,14,13,15)(17,19,23,31)(18,24,27,40)(20,29,34,47)(21,30,22,37)(25,38,32,48)(26,39,36,28)(33,45,35,46)(41,43,42,44);(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)(18,25)(19,29)(21,35)(22,33)(23,34)(24,38)(26,42)(27,32)(28,43)(30,46)(31,47)(36,41)(37,45)(39,44)(40,48);(17,21,23,22)(18,26,27,36)(19,30,31,37)(20,33,34,35)(24,39,40,28)(25,41,32,42)(29,45,47,46)(38,43,48,44)

Polycyclic presentation:

2^9|b^2=h,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, 6260305);