#8393247

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

4632 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300443);