#8202451

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

4742 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,d^2=gh,e^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

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, 8109647);