#9054521

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

4287 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8961717);