#9020068

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

4229 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8927264);