#8533643

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, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4117 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 8440839);