#9057341

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

4310 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,13)(14,16)(19,25)(23,26)(27,31)(30,32)(33,34)(35,41)(36,44)(37,38)(39,42)(40,45)(43,46)(47,48);(2,7)(4,10)(8,15)(12,14)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32)(33,34)(35,39)(36,40)(37,38)(41,42)(43,46)(44,45)(47,48);(3,11)(4,10)(5,9)(8,15)(18,22)(19,25)(20,28)(30,32)(34,38)(35,42)(36,45)(46,47);(1,3,6,11)(2,8,7,15)(4,14,10,12)(5,13,9,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,35,37,42)(34,39,38,41)(36,43,45,48)(40,46,44,47);(1,4,6,10)(2,9,7,5)(3,12,11,14)(8,13,15,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,34)(35,41)(36,40)(37,38)(39,42)(43,47)(44,45)(46,48);(1,5,6,9)(2,10,7,4)(3,13,11,16)(8,12,15,14)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,36,37,45)(34,40,38,44)(35,43,42,48)(39,46,41,47)

Polycyclic presentation:

2^9|b^2=hi,d^2=gh,e^2=g,f^2=gi,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, 8964537);