#8898567

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

4097 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8805763);