#9206493

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

4121 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)(3,10)(4,7)(5,8)(6,9)(11,16)(12,13)(14,15)(17,18,22,23)(19,25,30,24)(20,26,27,21)(28,29,31,32)(33,34)(35,42)(36,44)(37,40)(38,39)(41,46)(43,47)(45,48);(1,3,6,8)(2,5,9,10)(4,11,12,15)(7,14,13,16)(18,23)(20,27)(24,25)(28,31)(35,37)(40,42)(43,45)(47,48);(4,12)(7,13)(11,15)(14,16)(17,19,22,30)(18,24,23,25)(20,28,27,31)(21,29,26,32)(33,35,39,37)(34,40,38,42)(36,45,46,43)(41,48,44,47);(1,4,6,12)(2,7,9,13)(3,11,8,15)(5,14,10,16)(17,19)(18,25)(20,31)(21,29)(22,30)(23,24)(26,32)(27,28)(33,36,39,46)(34,41,38,44)(35,43,37,45)(40,47,42,48);(1,2)(3,5)(4,13)(6,9)(7,12)(8,10)(11,16)(14,15)(17,20)(18,26)(19,28)(21,23)(22,27)(24,32)(25,29)(30,31)(33,37,39,35)(34,40,38,42)(36,43,46,45)(41,48,44,47);(1,5)(2,8)(3,9)(4,14)(6,10)(7,15)(11,13)(12,16)(17,21,22,26)(18,27,23,20)(19,29,30,32)(24,31,25,28)(33,38,39,34)(35,42,37,40)(36,44,46,41)(43,48,45,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,c^2=h,d^2=gh,e^2=i,f^2=h,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, 9113689);