#8912289

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
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) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8819485);