#6351179

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351101
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

2740 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)(3,12)(4,9)(5,10)(6,15)(7,11)(8,13)(14,16)(17,18)(19,29)(20,26)(21,24)(22,28)(23,27)(25,32)(30,45)(31,41)(33,38)(34,48)(35,44)(36,39)(37,43)(40,47)(42,46);(1,3,7,4)(2,8,11,16)(5,13,6,14)(9,15,12,10)(18,24)(21,22)(25,38)(26,39)(27,28)(29,41)(31,32)(36,37)(40,48)(42,45)(43,44)(46,47);(1,4,7,3)(2,9,11,12)(5,13,6,14)(8,15,16,10)(17,19,23,33)(18,25,27,41)(20,34,35,30)(21,32,22,31)(24,38,28,29)(26,42,44,40)(36,46,37,47)(39,45,43,48);(1,5,7,6)(2,10,11,15)(3,13,4,14)(8,9,16,12)(17,20,23,35)(18,26,27,44)(19,30,33,34)(21,36,22,37)(24,39,28,43)(25,40,41,42)(29,45,38,48)(31,46,32,47);(2,11)(8,16)(9,12)(10,15)(18,27)(20,35)(24,28)(25,41)(29,38)(30,34)(36,37)(46,47)

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258375);