#9017148

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	#51964	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4160 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=ghi,d^2=gi,e^2=hi,f^2=h,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, 8924344);