#8509511

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

4486 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8416707);