#8977351

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

4069 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8884547);