#9054514

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

4307 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)(17,18,23,20)(19,25,27,21)(22,28,30,24)(26,29,32,31)(33,34,39,43)(35,44,48,40)(36,41,46,38)(37,42,45,47);(1,3,7,4)(2,8,5,10)(6,11,14,12)(9,15,13,16)(17,19,23,27)(18,21,20,25)(22,26,30,32)(24,29,28,31)(33,35,39,48)(34,40,43,44)(36,45,46,37)(38,47,41,42);(2,5)(3,4)(6,14)(15,16)(18,20)(21,25)(22,30)(26,32)(36,46)(37,45)(38,41)(42,47);(1,4,7,3)(2,8,5,10)(6,12,14,11)(9,15,13,16)(17,20,23,18)(19,25,27,21)(22,28,30,24)(26,31,32,29)(33,36)(34,41)(35,45)(37,48)(38,43)(39,46)(40,42)(44,47);(1,5,7,2)(3,10,4,8)(6,13,14,9)(11,16,12,15)(17,21,23,25)(18,19,20,27)(22,29,30,31)(24,26,28,32)(33,37,39,45)(34,42,43,47)(35,46,48,36)(38,40,41,44);(1,6)(2,9)(3,11)(4,12)(5,13)(7,14)(8,15)(10,16)(17,22,23,30)(18,24,20,28)(19,26,27,32)(21,29,25,31)(33,38,39,41)(34,36,43,46)(35,47,48,42)(37,40,45,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,d^2=h,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, 8961710);