#9726323

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

4100 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)(3,8)(4,13)(5,10)(6,14)(7,9)(11,16)(12,15)(17,18,22,20)(19,25,27,23)(21,24,30,28)(26,32,29,31)(36,44)(40,47)(43,45)(46,48);(1,3,7,4)(2,8,9,13)(5,14,15,11)(6,12,16,10)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(33,34)(35,42)(36,40)(37,39)(38,41)(43,48)(44,47)(45,46);(2,9)(3,4)(10,12)(11,14)(18,20)(23,25)(24,28)(31,32)(33,35,38,37)(34,39,41,42)(36,43,44,45)(40,46,47,48);(1,4,7,3)(2,8,9,13)(5,14,15,11)(6,10,16,12)(17,20,22,18)(19,25,27,23)(21,28,30,24)(26,32,29,31)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48);(1,5,7,15)(2,10,9,12)(3,11,4,14)(6,8,16,13)(17,21,22,30)(18,24,20,28)(19,26,27,29)(23,31,25,32)(33,37,38,35)(34,39,41,42)(36,45,44,43)(40,46,47,48);(1,6,7,16)(2,11,9,14)(3,12,4,10)(5,13,15,8)(17,19)(18,23)(20,25)(21,29)(22,27)(24,32)(26,30)(28,31)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=ghi,d^2=h,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,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, 9633519);