#9184717

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

4126 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)(3,8,11,14)(4,9,13,16)(5,12,15,10)(17,18)(19,24)(20,23)(21,25)(22,28)(26,31)(27,29)(30,32)(33,34,38,42)(35,39,45,40)(36,37,41,47)(43,44,46,48);(2,7)(4,13)(8,14)(10,12)(17,19,23,25)(18,24,20,21)(22,27,31,32)(26,30,28,29)(33,35)(34,39)(36,43)(37,44)(38,45)(40,42)(41,46)(47,48);(2,7)(3,11)(9,16)(10,12)(18,20)(19,25)(26,28)(27,32)(33,35,38,45)(34,39,42,40)(36,43,41,46)(37,44,47,48);(1,3)(2,8)(4,10)(5,9)(6,11)(7,14)(12,13)(15,16)(17,20,23,18)(19,21,25,24)(22,28,31,26)(27,29,32,30)(33,35)(34,40)(36,46)(37,44)(38,45)(39,42)(41,43)(47,48);(1,4,6,13)(2,9,7,16)(3,12,11,10)(5,8,15,14)(17,21)(18,25)(19,20)(22,30)(23,24)(26,27)(28,32)(29,31)(33,36)(34,37)(35,43)(38,41)(39,44)(40,48)(42,47)(45,46);(1,5,6,15)(2,10,7,12)(3,9,11,16)(4,14,13,8)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,30)(25,32)(33,37)(34,41)(35,44)(36,42)(38,47)(39,46)(40,43)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9091913);