#8139500

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

4049 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8046696);