#8757034

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

4115 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=gh,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8664230);