#9724820

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

4295 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)(3,7)(4,5)(6,8)(9,13)(10,11)(12,14)(15,16)(17,18)(19,28)(20,21)(22,32)(23,25)(24,29)(26,31)(27,30)(33,34)(35,45)(36,41)(37,42)(38,43)(39,44)(40,47)(46,48);(1,3,6,12)(2,7,8,14)(4,10,9,15)(5,11,13,16)(17,19,23,29)(18,24,25,28)(20,30,31,22)(21,32,26,27)(33,35)(34,40)(36,38)(37,46)(39,47)(41,48)(42,43)(44,45);(2,8)(4,9)(7,14)(10,15)(18,25)(19,29)(20,31)(27,32)(33,36,39,37)(34,41,44,42)(35,38,47,46)(40,48,45,43);(1,4,6,9)(2,5,8,13)(3,10,12,15)(7,11,14,16)(17,20)(18,26)(19,30)(21,25)(22,29)(23,31)(24,27)(28,32)(33,37,39,36)(34,42,44,41)(35,46,47,38)(40,43,45,48);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,21,23,26)(18,20,25,31)(19,27,29,32)(22,28,30,24)(33,38,39,46)(34,43,44,48)(35,36,47,37)(40,42,45,41);(1,5)(2,9)(3,11)(4,8)(6,13)(7,15)(10,14)(12,16)(17,22,23,30)(18,27,25,32)(19,20,29,31)(21,28,26,24)(33,34)(35,40)(36,41)(37,42)(38,48)(39,44)(43,46)(45,47)

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=gh,e^2=g,f^2=i,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, 9632016);