#8359929

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

4142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,d^2=ghi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8267125);