#8358682

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

4400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=gi,e^2=gh,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, 8265878);