#9054511

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

4303 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,18,23,21)(19,24,20,26)(22,29,30,25)(27,32,28,31)(33,34)(35,42)(36,40)(37,41)(38,39)(43,46)(44,48)(45,47);(2,7)(4,8)(10,14)(12,15)(17,19,23,20)(18,24,21,26)(22,27,30,28)(25,31,29,32)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(3,9)(4,8)(5,13)(10,14)(18,21)(19,20)(22,30)(31,32)(34,38)(36,42)(37,45)(43,48);(1,3)(2,8)(4,7)(5,11)(6,9)(10,15)(12,14)(13,16)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32)(33,36,39,42)(34,40,38,35)(37,46,45,44)(41,43,47,48);(1,4,6,8)(2,9,7,3)(5,12,13,15)(10,16,14,11)(17,21,23,18)(19,26,20,24)(22,29,30,25)(27,32,28,31)(33,37,39,45)(34,41,38,47)(35,43,40,48)(36,44,42,46);(1,5,6,13)(2,10,7,14)(3,11,9,16)(4,12,8,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,38,39,34)(35,42,40,36)(37,47,45,41)(43,46,48,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8961707);