#8228703

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

4126 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:

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

Polycyclic presentation:

2^9|b^2=gi,d^2=gi,e^2=i,f^2=gh,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, 8135899);