#9160801

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4016 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)(3,7)(4,12)(5,6)(8,14)(9,11)(10,16)(13,15)(17,18,23,25)(19,28,22,24)(20,29,30,26)(21,27,32,31)(33,34)(35,41)(36,39)(37,42)(38,44)(40,45)(43,47)(46,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,19,23,22)(18,24,25,28)(20,27,30,31)(21,26,32,29)(33,35)(34,38)(36,43)(37,44)(39,46)(40,47)(41,42)(45,48);(2,6)(7,9)(8,12)(15,16)(18,25)(20,30)(24,28)(27,31)(33,36,37,45)(34,39,42,40)(35,43,44,48)(38,46,41,47);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,14)(13,15)(17,20,23,30)(18,26,25,29)(19,27,22,31)(21,24,32,28)(33,36)(34,40)(35,43)(37,45)(38,47)(39,42)(41,46)(44,48);(1,3,5,11)(2,7,6,9)(4,13,14,10)(8,16,12,15)(17,21,23,32)(18,27,25,31)(19,29,22,26)(20,28,30,24)(33,34)(35,38)(36,39)(37,42)(40,45)(41,44)(43,46)(47,48);(1,4,5,14)(2,8,6,12)(3,10,11,13)(7,15,9,16)(17,22,23,19)(18,28,25,24)(20,31,30,27)(21,29,32,26)(33,35)(34,41)(36,43)(37,44)(38,42)(39,47)(40,46)(45,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=ghi,d^2=hi,e^2=h,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^d=egi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9067997);