#9057346

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
prime radical	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 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)(19,23)(20,24)(21,22)(25,27)(26,31)(28,32)(29,30)(33,34,39,40)(35,43,44,41)(36,46,45,42)(37,38,47,48);(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(18,22)(23,25)(24,30)(31,32)(34,40)(35,44)(37,47)(42,46);(2,6)(3,9)(4,12)(14,16)(17,19,21,27)(18,23,22,25)(20,28,29,26)(24,32,30,31)(33,35,39,44)(34,41,40,43)(36,37,45,47)(38,46,48,42);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,15)(13,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,29)(30,32)(33,36,39,45)(34,42,40,46)(35,37,44,47)(38,43,48,41);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,31)(29,30)(33,37)(34,38)(35,45)(36,44)(39,47)(40,48)(41,46)(42,43);(1,4,5,12)(2,8,6,13)(3,11,9,15)(7,14,10,16)(17,20,21,29)(18,24,22,30)(19,26,27,28)(23,31,25,32)(33,38)(34,37)(35,46)(36,43)(39,48)(40,47)(41,45)(42,44)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=g,d^2=h,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, 8964542);