#9066074

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	#41140	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4054 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,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, 8973270);