#8931066

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

4140 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)(3,8,11,15)(4,9,13,12)(5,10,16,14)(17,18)(19,24)(20,21)(22,30)(23,25)(26,31)(27,29)(28,32)(35,41)(39,46)(42,44)(47,48);(2,7)(4,13)(8,15)(10,14)(17,19,23,20)(18,24,25,21)(22,27,31,28)(26,32,30,29)(33,34)(35,39)(36,43)(37,38)(40,45)(41,46)(42,48)(44,47);(3,11)(5,16)(8,15)(10,14)(18,25)(19,20)(22,31)(29,32)(34,38)(35,41)(40,43)(42,44);(1,3,6,11)(2,8,7,15)(4,14,13,10)(5,12,16,9)(17,20,23,19)(18,24,25,21)(22,28,31,27)(26,32,30,29)(33,35)(34,39)(36,44)(37,41)(38,46)(40,48)(42,45)(43,47);(1,4)(2,9)(3,10)(5,15)(6,13)(7,12)(8,16)(11,14)(17,21,23,24)(18,20,25,19)(22,29,31,32)(26,28,30,27)(33,36,37,45)(34,40,38,43)(35,42,41,44)(39,47,46,48);(1,5,6,16)(2,10,7,14)(3,12,11,9)(4,15,13,8)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,32)(25,30)(33,34)(35,39)(36,40)(37,38)(41,46)(42,47)(43,45)(44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8838262);