#9066076

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	#48082	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4231 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=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, 8973272);