#9057338

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

4298 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=h,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, 8964534);