#8334344

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

4206 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=i,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=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, 8241540);