#8722269

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3941 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,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,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, 8629465);