#8649875

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3821 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=ghi,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8557071);