#8757015

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

4131 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)(19,24)(20,22)(21,30)(23,25)(26,29)(27,32)(28,31)(35,41)(38,42)(43,47)(46,48);(2,5)(4,13)(7,9)(11,16)(17,19)(18,24)(20,29)(21,27)(22,26)(23,28)(25,31)(30,32)(33,34)(35,42)(36,44)(37,40)(38,41)(39,45)(43,46)(47,48);(3,10)(7,9)(11,16)(14,15)(18,25)(19,28)(20,27)(26,30)(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)(17,20,23,27)(18,22,25,32)(19,29,28,21)(24,26,31,30)(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)(17,21,23,29)(18,26,25,30)(19,27,28,20)(22,31,32,24)(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)(17,22)(18,27)(19,30)(20,25)(21,31)(23,32)(24,29)(26,28)(33,36)(34,39)(35,43)(37,45)(38,46)(40,44)(41,47)(42,48)

Polycyclic presentation:

2^9|a^2=g,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, 8664211);