#9027089

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	#41182	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3998 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8934285);