#8720821

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	#53410	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4010 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)(3,4)(5,9)(6,7)(8,10)(11,12)(13,14)(15,16)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(33,34)(35,43)(36,38)(37,41)(39,40)(42,46)(44,45)(47,48);(1,2)(3,10)(4,8)(5,9)(6,7)(11,14)(12,13)(15,16)(17,19,22,27)(18,23,21,25)(20,26,29,32)(24,30,28,31)(34,40)(36,46)(41,43)(44,47);(2,7)(3,8)(9,16)(11,13)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,35,39,37)(34,41,40,43)(36,47,46,44)(38,45,42,48);(1,3,6,8)(2,4,7,10)(5,13,15,11)(9,14,16,12)(17,19)(18,23)(20,26)(21,25)(22,27)(24,30)(28,31)(29,32)(33,36,39,46)(34,42,40,38)(35,44,37,47)(41,45,43,48);(1,4)(2,8)(3,7)(5,14)(6,10)(9,11)(12,15)(13,16)(17,20,22,29)(18,24,21,28)(19,26,27,32)(23,30,25,31)(33,37,39,35)(34,43,40,41)(36,47,46,44)(38,45,42,48);(1,5,6,15)(2,9,7,16)(3,11,8,13)(4,12,10,14)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,31,32,30)(33,38,39,42)(34,36,40,46)(35,45,37,48)(41,47,43,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=hi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8628017);