#8202482

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

4071 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)(4,12,14,8)(10,15,13,16)(18,22)(19,25)(24,29)(26,31)(33,34,39,42)(35,40,46,41)(36,47,37,38)(43,45,44,48);(1,3,6,11)(2,7,5,9)(4,13,14,10)(8,16,12,15)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,39,46)(34,40,42,41)(36,44,37,43)(38,45,47,48);(2,5)(7,9)(8,12)(15,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36,39,37)(34,38,42,47)(35,43,46,44)(40,48,41,45);(1,3)(2,7)(4,10)(5,9)(6,11)(8,15)(12,16)(13,14)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,37,39,36)(34,38,42,47)(35,44,46,43)(40,48,41,45);(1,4,6,14)(2,8,5,12)(3,10,11,13)(7,15,9,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,35)(34,41)(36,43)(37,44)(38,45)(39,46)(40,42)(47,48);(1,5,6,2)(3,9,11,7)(4,12,14,8)(10,16,13,15)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,38)(34,36)(35,45)(37,42)(39,47)(40,44)(41,43)(46,48)

Polycyclic presentation:

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

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, 8109678);