#8720378

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

4095 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)(3,4,10,9)(5,8,15,16)(11,12,13,14)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,31,30,32)(33,34)(35,38)(36,46)(37,48)(39,43)(40,45)(41,47)(42,44);(1,2,6,7)(3,9,10,4)(5,8,15,16)(11,14,13,12)(17,19)(18,23)(20,26)(21,25)(22,27)(24,31)(28,32)(29,30)(33,35)(34,40)(36,37)(38,43)(39,45)(41,48)(42,46)(44,47);(2,7)(3,10)(8,16)(11,13)(20,29)(24,28)(26,30)(31,32)(33,36,39,47)(34,41,43,46)(35,37,45,44)(38,42,40,48);(1,3)(2,4)(5,13)(6,10)(7,9)(8,14)(11,15)(12,16)(17,20,22,29)(18,24,21,28)(19,26,27,30)(23,31,25,32)(33,36)(34,41)(35,44)(37,45)(38,48)(39,47)(40,42)(43,46);(1,4)(2,3)(5,14)(6,9)(7,10)(8,13)(11,16)(12,15)(17,21,22,18)(19,25,27,23)(20,28,29,24)(26,32,30,31)(33,37,39,44)(34,42,43,48)(35,36,45,47)(38,41,40,46);(1,5,6,15)(2,8,7,16)(3,11,10,13)(4,12,9,14)(17,19)(18,25)(20,30)(21,23)(22,27)(24,31)(26,29)(28,32)(33,38,39,40)(34,35,43,45)(36,42,47,48)(37,46,44,41)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=ghi,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, 8627574);