#9015624

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

4149 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=hi,e^2=i,f^2=gh,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, 8922820);