#9010559

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

4039 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=ghi,e^2=gi,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, 8917755);