#8456003

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

4540 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=hi,d^2=g,e^2=gi,f^2=g,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, 8363199);