#9230871

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	#50951	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4128 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)(3,9)(4,6)(5,8)(7,11)(10,16)(12,14)(13,15)(17,18,22,26)(19,27,30,23)(20,24,29,31)(21,32,28,25)(33,34)(35,36)(37,45)(38,41)(39,43)(40,48)(42,46)(44,47);(1,3)(2,7)(4,9)(5,10)(6,11)(8,15)(12,16)(13,14)(17,19,22,30)(18,23,26,27)(20,21,29,28)(24,25,31,32)(33,35,38,43)(34,36,41,39)(37,42,48,47)(40,44,45,46);(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,32)(29,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,45)(47,48);(1,4,6,2)(3,9,11,7)(5,8,14,12)(10,15,13,16)(17,18)(19,23)(20,31)(21,32)(22,26)(24,29)(25,28)(27,30)(33,36,38,39)(34,35,41,43)(37,46,48,44)(40,47,45,42);(1,5,6,14)(2,8,4,12)(3,10,11,13)(7,15,9,16)(17,20)(18,24)(19,28)(21,30)(22,29)(23,32)(25,27)(26,31)(33,35)(34,36)(37,47)(38,43)(39,41)(40,46)(42,48)(44,45);(1,3)(2,9)(4,7)(5,13)(6,11)(8,15)(10,14)(12,16)(17,21,22,28)(18,25,26,32)(19,29,30,20)(23,31,27,24)(33,37,38,48)(34,40,41,45)(35,42,43,47)(36,44,39,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=gh,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,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, 9138067);