#8256610

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

4054 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:

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

Polycyclic presentation:

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

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, 8163806);