#9726387

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

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=gh,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, 9633583);