#9127594

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	#40244	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4151 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)(3,6)(4,9)(5,10)(7,8)(11,16)(12,14)(13,15)(17,18,23,27)(19,24,28,31)(20,25,30,29)(21,22,26,32)(33,34)(35,42)(36,40)(37,45)(38,39)(41,47)(43,44)(46,48);(1,3,7,11)(2,6,8,16)(4,12,13,5)(9,14,15,10)(17,19)(18,24)(20,22)(21,29)(23,28)(25,26)(27,31)(30,32)(34,39)(36,42)(41,45)(44,46);(2,8)(3,11)(4,13)(10,14)(19,28)(20,30)(24,31)(25,29)(33,35,38,40)(34,36,39,42)(37,43,47,48)(41,44,45,46);(1,4)(2,9)(3,12)(5,11)(6,14)(7,13)(8,15)(10,16)(17,20)(18,25)(19,22)(21,31)(23,30)(24,26)(27,29)(28,32)(33,35)(34,36)(37,43)(38,40)(39,42)(41,44)(45,46)(47,48);(1,5,7,12)(2,10,8,14)(3,13,11,4)(6,15,16,9)(17,21)(18,22)(19,29)(20,24)(23,26)(25,28)(27,32)(30,31)(33,36)(34,40)(35,39)(37,46)(38,42)(41,43)(44,47)(45,48);(1,6)(2,11)(3,8)(4,14)(5,9)(7,16)(10,13)(12,15)(17,22)(18,26)(19,20)(21,27)(23,32)(24,25)(28,30)(29,31)(33,37,38,47)(34,41,39,45)(35,43,40,48)(36,44,42,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,e^2=gh,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, 9034790);