#8720733

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

4206 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,d^2=i,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8627929);