#8114370

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

4123 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)(3,9)(4,12)(5,8)(6,10)(7,13)(11,14)(15,16)(18,22)(19,25)(20,28)(30,32)(34,37)(35,36)(42,45)(43,44);(1,2,5,8)(3,6,10,9)(4,7,13,12)(11,14,16,15)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,34,39,37)(35,41,36,40)(38,42,46,45)(43,48,44,47);(3,10)(6,9)(11,16)(14,15)(17,18,21,22)(19,23,25,26)(20,24,28,29)(27,30,31,32)(33,35,39,36)(34,40,37,41)(38,43,46,44)(42,47,45,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,26)(27,30)(28,29)(31,32)(33,36,39,35)(34,41,37,40)(38,44,46,43)(42,48,45,47);(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,27)(21,25)(22,26)(24,30)(28,31)(29,32)(33,37,39,34)(35,41,36,40)(38,45,46,42)(43,48,44,47);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,38,39,46)(34,42,37,45)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8021566);