#9040453

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

4519 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=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, 8947649);