#8746071

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

3943 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=hi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8653267);