#8726386

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	#40170	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4086 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,e^2=ghi,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, 8633582);