#8358271

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

4420 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=g,d^2=hi,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, 8265467);