#8720607

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

4100 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=i,d^2=gi,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, 8627803);