#9498412

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	#57957	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3846 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=hi,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9405608);