#8992911

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

4230 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=i,d^2=ghi,e^2=h,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, 8900107);