#8114409

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

4376 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)(9,10,14,12)(11,17,18,15)(13,20,21,16)(19,22,24,23)(25,26,30,31)(27,34,33,29)(28,36,37,32)(35,38,39,40);(2,5)(6,7)(10,12)(15,17)(16,20)(22,23)(26,31)(27,33)(28,37)(38,40);(1,2)(3,6)(4,5)(7,8)(11,18)(15,17)(19,24)(22,23)(26,31)(27,33)(32,36)(35,39);(1,2)(3,6)(4,5)(7,8)(9,11,14,18)(10,15,12,17)(13,19,21,24)(16,22,20,23)(25,27,30,33)(26,29,31,34)(28,35,37,39)(32,38,36,40);(1,3)(2,6)(4,8)(5,7)(9,12,14,10)(11,17,18,15)(13,20,21,16)(19,23,24,22)(25,28,30,37)(26,32,31,36)(27,35,33,39)(29,38,34,40);(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,29)(26,33)(27,31)(28,38)(30,34)(32,39)(35,36)(37,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8021605);