#8630930

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

4158 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=gi,e^2=hi,f^2=g,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, 8538126);