#9028541

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

4604 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8935737);