#6357618

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357552
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1819 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264814);