#8630895

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

4149 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:

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

Polycyclic presentation:

2^9|a^2=i,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, 8538091);