#9724821

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

4103 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)(19,29)(20,23)(21,26)(22,27)(24,28)(25,32)(30,31)(33,34)(35,38)(36,42)(37,47)(39,41)(40,43)(44,48)(45,46);(1,3,6,12)(2,7,8,14)(4,10,9,15)(5,11,13,16)(17,19)(18,25)(20,22)(21,30)(23,31)(24,32)(26,27)(28,29)(33,35,40,37)(34,38,43,47)(36,45,39,44)(41,48,42,46);(2,8)(4,9)(7,14)(10,15)(17,20,24,21)(18,23,28,26)(19,22,32,30)(25,31,29,27)(33,36,40,39)(34,41,43,42)(35,44,37,45)(38,46,47,48);(1,4,6,9)(2,5,8,13)(3,10,12,15)(7,11,14,16)(17,21,24,20)(18,26,28,23)(19,30,32,22)(25,27,29,31)(33,37,40,35)(34,38,43,47)(36,44,39,45)(41,48,42,46);(1,3)(2,7)(4,10)(5,11)(6,12)(8,14)(9,15)(13,16)(17,22,24,30)(18,27,28,31)(19,20,32,21)(23,25,26,29)(33,38)(34,35)(36,46)(37,43)(39,48)(40,47)(41,44)(42,45);(1,5)(2,9)(3,11)(4,8)(6,13)(7,15)(10,14)(12,16)(17,23,24,26)(18,20,28,21)(19,31,32,27)(22,29,30,25)(33,39,40,36)(34,42,43,41)(35,44,37,45)(38,46,47,48)

Polycyclic presentation:

2^9|b^2=h,c^2=gi,d^2=gh,e^2=g,f^2=gi,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, 9632017);