#9034468

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

4290 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)(3,6)(4,13)(5,9)(7,10)(8,12)(11,16)(14,15)(17,18,23,20)(19,27,28,24)(21,25,32,29)(22,26,31,30)(33,34,39,41)(35,42,36,40)(37,38,47,48)(43,46,45,44);(1,3)(2,6)(4,5)(7,11)(8,14)(9,13)(10,16)(12,15)(17,19,23,28)(18,24,20,27)(21,31,32,22)(25,26,29,30)(33,35,39,36)(34,40,41,42)(37,43,47,45)(38,44,48,46);(3,11)(5,15)(6,16)(9,14)(21,32)(22,31)(25,29)(26,30)(33,35,39,36)(34,40,41,42)(37,45,47,43)(38,46,48,44);(1,4)(2,8)(3,5)(6,14)(7,12)(9,16)(10,13)(11,15)(17,20,23,18)(19,27,28,24)(21,25,32,29)(22,30,31,26)(33,36,39,35)(34,40,41,42)(37,43,47,45)(38,46,48,44);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,14)(13,16)(17,21)(18,25)(19,22)(20,29)(23,32)(24,30)(26,27)(28,31)(33,37)(34,38)(35,43)(36,45)(39,47)(40,44)(41,48)(42,46);(1,6)(2,3)(4,14)(5,8)(7,16)(9,12)(10,11)(13,15)(17,22)(18,26)(19,21)(20,30)(23,31)(24,29)(25,27)(28,32)(33,38)(34,37)(35,44)(36,46)(39,48)(40,43)(41,47)(42,45)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8941664);