#9717541

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

4049 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18)(19,24)(20,25)(21,29)(22,23)(26,31)(27,30)(28,32)(33,34)(35,42)(36,40)(37,45)(38,39)(41,47)(43,44)(46,48);(4,11)(8,14)(10,15)(12,16)(18,23)(20,28)(24,30)(26,29)(33,34)(35,36)(37,41)(38,39)(40,42)(43,44)(45,47)(46,48);(2,7)(3,9)(8,14)(10,15)(18,23)(19,27)(25,32)(26,29)(34,39)(35,40)(41,45)(43,48);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,19,22,27)(18,24,23,30)(20,26,28,29)(21,25,31,32)(33,35,38,40)(34,36,39,42)(37,43,47,48)(41,44,45,46);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,20)(18,25)(19,26)(21,30)(22,28)(23,32)(24,31)(27,29)(33,36)(34,40)(35,39)(37,46)(38,42)(41,43)(44,47)(45,48);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,21,22,31)(18,26,23,29)(19,25,27,32)(20,24,28,30)(33,37)(34,41)(35,43)(36,44)(38,47)(39,45)(40,48)(42,46)

Polycyclic presentation:

2^9|d^2=h,e^2=ghi,f^2=g,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, 9624737);