#9054483

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

4166 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=gh,e^2=i,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, 8961679);