#9054520

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

4287 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=gh,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, 8961716);