#9016603

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

4274 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=hi,d^2=ghi,e^2=i,f^2=gh,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, 8923799);