#9726603

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

4297 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)(3,5)(4,11)(6,14)(7,8)(9,16)(10,15)(12,13)(17,18)(19,20)(21,29)(22,24)(23,27)(25,31)(26,32)(28,30)(33,34)(35,42)(36,38)(37,41)(39,43)(40,47)(44,48)(45,46);(1,3,7,4)(2,5,8,11)(6,10,16,12)(9,13,14,15)(17,19,22,27)(18,23,24,20)(21,26,31,30)(25,32,29,28)(33,35)(34,40)(36,44)(37,45)(38,46)(39,47)(41,48)(42,43);(2,8)(3,4)(9,14)(10,12)(18,24)(20,23)(25,29)(28,32)(33,36,39,37)(34,38,43,41)(35,44,47,45)(40,46,42,48);(1,4,7,3)(2,5,8,11)(6,12,16,10)(9,13,14,15)(17,20)(18,19)(21,28)(22,23)(24,27)(25,26)(29,30)(31,32)(33,37,39,36)(34,41,43,38)(35,45,47,44)(40,48,42,46);(1,5)(2,3)(4,8)(6,15)(7,11)(9,12)(10,14)(13,16)(17,19)(18,20)(21,30)(22,27)(23,24)(25,32)(26,31)(28,29)(33,35)(34,42)(36,44)(37,45)(38,48)(39,47)(40,43)(41,46);(1,6,7,16)(2,9,8,14)(3,10,4,12)(5,13,11,15)(17,21)(18,25)(19,26)(20,28)(22,31)(23,32)(24,29)(27,30)(33,38,39,41)(34,36,43,37)(35,46,47,48)(40,44,42,45)

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=gh,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, 9633799);