#9039457

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

4111 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)(17,18,23,27)(19,28,20,24)(21,25,26,22)(29,30,31,32)(33,34,38,42)(35,40,45,39)(36,41,47,37)(43,44,46,48);(1,2,5,6)(3,7,9,10)(4,8,12,13)(11,14,15,16)(17,19,23,20)(18,24,27,28)(21,29,26,31)(22,30,25,32)(33,35,38,45)(34,39,42,40)(36,43,47,46)(37,44,41,48);(2,6)(4,12)(7,10)(11,15)(17,19,23,20)(18,24,27,28)(21,31,26,29)(22,32,25,30)(33,35,38,45)(34,40,42,39)(36,43,47,46)(37,48,41,44);(1,3,5,9)(2,7,6,10)(4,11,12,15)(8,14,13,16)(17,20,23,19)(18,24,27,28)(21,29,26,31)(22,32,25,30)(33,36)(34,41)(35,43)(37,42)(38,47)(39,48)(40,44)(45,46);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,21,23,26)(18,25,27,22)(19,29,20,31)(24,32,28,30)(33,35)(34,40)(36,46)(37,44)(38,45)(39,42)(41,48)(43,47);(1,4)(2,8)(3,11)(5,12)(6,13)(7,14)(9,15)(10,16)(17,22,23,25)(18,26,27,21)(19,30,20,32)(24,31,28,29)(33,37,38,41)(34,36,42,47)(35,44,45,48)(39,43,40,46)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8946653);