#9057322

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

4148 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,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, 8964518);