#8630909

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

4164 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)(10,14,16,15)(17,18,22,25)(19,21,24,31)(20,27,30,23)(26,32,28,29)(34,39)(35,42)(36,44)(46,47);(1,3,6,11)(2,7,5,9)(4,10,13,16)(8,14,12,15)(17,19,22,24)(18,21,25,31)(20,28,30,26)(23,32,27,29)(33,34)(35,37)(36,45)(38,39)(40,44)(41,42)(43,46)(47,48);(1,3)(2,7)(4,10)(5,9)(6,11)(8,14)(12,15)(13,16)(17,20,22,30)(18,23,25,27)(19,26,24,28)(21,29,31,32)(33,35,38,42)(34,37,39,41)(36,43,44,48)(40,47,45,46);(1,3)(2,7)(4,10)(5,9)(6,11)(8,14)(12,15)(13,16)(17,19)(18,21)(20,26)(22,24)(23,29)(25,31)(27,32)(28,30)(33,36)(34,40)(35,43)(37,47)(38,44)(39,45)(41,46)(42,48);(1,4,6,13)(2,8,5,12)(3,10,11,16)(7,14,9,15)(17,20)(18,23)(19,26)(21,29)(22,30)(24,28)(25,27)(31,32)(33,35)(34,41)(36,43)(37,39)(38,42)(40,46)(44,48)(45,47);(1,5,6,2)(3,9,11,7)(4,12,13,8)(10,15,16,14)(17,21)(18,24)(19,25)(20,29)(22,31)(23,28)(26,27)(30,32)(33,37,38,41)(34,35,39,42)(36,46,44,47)(40,48,45,43)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,e^2=ghi,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, 8538105);