#9712427

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

4061 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)(3,6)(4,5)(7,8)(9,13)(10,16)(11,14)(12,15)(17,18,23,27)(19,26,22,24)(20,25,31,21)(28,32,30,29)(33,34,39,37)(35,43,45,40)(36,46,47,41)(38,48,44,42);(1,3,7,10)(2,6,8,16)(4,11,13,15)(5,14,9,12)(17,19,23,22)(18,24,27,26)(20,28,31,30)(21,32,25,29)(33,35,39,45)(34,40,37,43)(36,44,47,38)(41,42,46,48);(2,8)(3,10)(4,13)(12,14)(17,20,23,31)(18,21,27,25)(19,28,22,30)(24,32,26,29)(33,36,39,47)(34,41,37,46)(35,38,45,44)(40,48,43,42);(1,4)(2,9)(3,11)(5,8)(6,12)(7,13)(10,15)(14,16)(17,20)(18,25)(19,28)(21,27)(22,30)(23,31)(24,29)(26,32)(33,36)(34,41)(35,44)(37,46)(38,45)(39,47)(40,42)(43,48);(1,5,7,9)(2,4,8,13)(3,12,10,14)(6,15,16,11)(17,21,23,25)(18,20,27,31)(19,29,22,32)(24,30,26,28)(33,37,39,34)(35,40,45,43)(36,46,47,41)(38,48,44,42);(1,6)(2,10)(3,8)(4,12)(5,11)(7,16)(9,15)(13,14)(17,22,23,19)(18,26,27,24)(20,30,31,28)(21,29,25,32)(33,38,39,44)(34,42,37,48)(35,36,45,47)(40,46,43,41)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,e^2=h,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, 9619623);