#9230889

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	#40126	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4156 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:

(2,3)(7,9)(8,10)(15,16)(17,18,22,23)(19,27,29,24)(20,25,30,31)(21,28,32,26)(33,34)(35,36)(37,45)(38,41)(39,43)(40,48)(42,46)(44,47);(1,2,6,3)(4,7,12,9)(5,8,14,10)(11,15,13,16)(18,23)(20,30)(24,27)(26,28)(33,35,38,43)(34,36,41,39)(37,42,48,47)(40,44,45,46);(17,19)(18,24)(20,28)(21,25)(22,29)(23,27)(26,30)(31,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,45)(47,48);(1,3,6,2)(4,7,12,9)(5,8,14,10)(11,16,13,15)(17,19)(18,24)(20,26)(21,31)(22,29)(23,27)(25,32)(28,30)(33,36,38,39)(34,35,41,43)(37,46,48,44)(40,47,45,42);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,20)(18,25)(19,28)(21,24)(22,30)(23,31)(26,29)(27,32)(33,35)(34,36)(37,47)(38,43)(39,41)(40,46)(42,48)(44,45);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21)(18,26)(19,25)(20,27)(22,32)(23,28)(24,30)(29,31)(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=gh,d^2=gh,e^2=ghi,f^2=gh,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, 9138085);