#9159911

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	#40723	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4191 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)(5,7)(6,9)(8,13)(11,15)(14,16)(17,18,22,23)(19,24,27,30)(20,25,31,28)(21,29,32,26)(34,37)(35,41)(40,44)(42,45);(1,3,6,5)(2,7,9,10)(4,11,13,14)(8,15,12,16)(17,18,22,23)(19,24,27,30)(20,28,31,25)(21,26,32,29)(33,34,38,37)(35,39,41,43)(36,44,46,40)(42,48,45,47);(4,13)(8,12)(11,14)(15,16)(18,23)(19,27)(25,28)(26,29)(33,35,38,41)(34,39,37,43)(36,42,46,45)(40,47,44,48);(1,4,6,13)(2,8,9,12)(3,11,5,14)(7,15,10,16)(17,19)(18,24)(20,29)(21,28)(22,27)(23,30)(25,32)(26,31)(33,35)(34,39)(36,45)(37,43)(38,41)(40,48)(42,46)(44,47);(1,2)(3,7)(4,12)(5,10)(6,9)(8,13)(11,16)(14,15)(17,20)(18,25)(19,26)(21,24)(22,31)(23,28)(27,29)(30,32)(33,36)(34,40)(35,42)(37,44)(38,46)(39,47)(41,45)(43,48);(1,5,6,3)(2,7,9,10)(4,14,13,11)(8,15,12,16)(17,21)(18,26)(19,28)(20,30)(22,32)(23,29)(24,31)(25,27)(33,37,38,34)(35,43,41,39)(36,40,46,44)(42,47,45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=hi,d^2=ghi,f^2=g,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, 9067107);