#9063294

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	#35405	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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)(3,10)(4,8)(5,9)(6,7)(11,16)(12,15)(13,14)(17,18,23,21)(19,26,25,22)(20,28,30,24)(27,31,29,32)(33,34,39,42)(35,40,43,36)(37,38,41,48)(44,47,46,45);(2,7)(3,9)(4,12)(14,16)(17,19)(18,22)(20,29)(21,26)(23,25)(24,32)(27,30)(28,31)(33,35)(34,40)(36,42)(37,46)(38,45)(39,43)(41,44)(47,48);(3,9)(4,12)(5,10)(8,15)(18,21)(20,30)(22,26)(27,29)(35,43)(36,40)(37,41)(38,48);(1,3,6,9)(2,5,7,10)(4,13,12,11)(8,16,15,14)(17,20,23,30)(18,24,21,28)(19,27,25,29)(22,31,26,32)(33,36)(34,35)(37,47)(38,46)(39,40)(41,45)(42,43)(44,48);(1,4,6,12)(2,8,7,15)(3,11,9,13)(5,14,10,16)(17,21,23,18)(19,26,25,22)(20,24,30,28)(27,31,29,32)(33,37,39,41)(34,38,42,48)(35,44,43,46)(36,45,40,47);(1,5)(2,9)(3,7)(4,14)(6,10)(8,13)(11,15)(12,16)(17,22,23,26)(18,25,21,19)(20,31,30,32)(24,29,28,27)(33,38)(34,41)(35,45)(36,46)(37,42)(39,48)(40,44)(43,47)

Polycyclic presentation:

2^9|a^2=hi,d^2=g,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8970490);