#8358529

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

4487 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,d^2=gi,e^2=h,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, 8265725);