#9057455

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3907 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:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=gh,d^2=g,e^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, 8964651);