#6350444

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350397
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2766 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257640);