#8463882

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

4445 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8371078);