#8824462

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

4065 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)(3,8,11,5)(4,9,14,15)(10,16,12,13)(17,18,23,20)(19,26,25,22)(21,24,29,28)(27,32,31,30)(33,34,39,36)(35,40,37,41)(38,42,46,44)(43,47,45,48);(2,7)(5,8)(9,15)(13,16)(21,29)(24,28)(27,31)(30,32)(33,35,39,37)(34,40,36,41)(38,45,46,43)(42,48,44,47);(1,3,6,11)(2,8,7,5)(4,12,14,10)(9,13,15,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,32)(29,31)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,3)(2,5)(4,10)(6,11)(7,8)(9,13)(12,14)(15,16)(17,20,23,18)(19,26,25,22)(21,28,29,24)(27,32,31,30)(33,35,39,37)(34,40,36,41)(38,43,46,45)(42,47,44,48);(1,4)(2,9)(3,10)(5,13)(6,14)(7,15)(8,16)(11,12)(17,21)(18,24)(19,27)(20,28)(22,30)(23,29)(25,31)(26,32)(33,37,39,35)(34,41,36,40)(38,45,46,43)(42,48,44,47);(1,5)(2,3)(4,13)(6,8)(7,11)(9,10)(12,15)(14,16)(17,22,23,26)(18,25,20,19)(21,30,29,32)(24,31,28,27)(33,38,39,46)(34,42,36,44)(35,43,37,45)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=gi,d^2=hi,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,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, 8731658);