#8115543

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

4568 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=hi,e^2=g,f^2=gi,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, 8022739);