#8639795

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
prime radical	#45649	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4084 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)(3,5)(4,8)(6,7)(9,10)(11,15)(12,13)(14,16)(17,18)(19,26)(20,28)(21,23)(22,24)(25,30)(27,31)(29,32)(33,34,39,37)(35,40,36,43)(38,41,46,45)(42,47,44,48);(1,3,6,10)(2,5,7,9)(4,12,14,11)(8,13,16,15)(17,19,23,22)(18,24,21,26)(20,29,30,27)(25,32,28,31)(34,37)(40,43)(41,45)(47,48);(2,7)(3,10)(8,16)(11,12)(18,21)(24,26)(25,28)(31,32)(33,35,39,36)(34,40,37,43)(38,42,46,44)(41,47,45,48);(1,4)(2,8)(3,11)(5,15)(6,14)(7,16)(9,13)(10,12)(17,20,23,30)(18,25,21,28)(19,27,22,29)(24,31,26,32)(33,36,39,35)(34,40,37,43)(38,42,46,44)(41,48,45,47);(1,3)(2,9)(4,11)(5,7)(6,10)(8,13)(12,14)(15,16)(17,21,23,18)(19,24,22,26)(20,28,30,25)(27,31,29,32)(33,37,39,34)(35,40,36,43)(38,45,46,41)(42,47,44,48);(1,5,6,9)(2,3,7,10)(4,13,14,15)(8,12,16,11)(17,22,23,19)(18,24,21,26)(20,27,30,29)(25,32,28,31)(33,38,39,46)(34,41,37,45)(35,42,36,44)(40,47,43,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=g,e^2=g,f^2=gi,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, 8546991);