#6350872

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350717
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

2583 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258068);