#8662956

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

4071 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8570152);