#8119313

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

4190 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8026509);