#8977263

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

4085 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=g,e^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, 8884459);