#9015500

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

4674 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)(3,8)(4,7)(5,6)(10,15)(11,17)(16,22)(18,23)(25,26,30,31)(27,35,38,32)(28,33,39,37)(29,36,40,34);(3,4)(7,8)(9,10)(11,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,24)(26,31)(28,39)(32,35)(34,36);(1,3)(2,7)(4,6)(5,8)(11,17)(12,19)(13,21)(16,22)(25,27,30,38)(26,32,31,35)(28,34,39,36)(29,33,40,37);(3,4)(7,8)(9,11,14,17)(10,12,15,19)(13,18,21,23)(16,20,22,24)(25,27)(26,32)(28,34)(29,37)(30,38)(31,35)(33,40)(36,39);(1,4,6,3)(2,7,5,8)(9,12)(10,11)(13,20)(14,19)(15,17)(16,18)(21,24)(22,23)(25,28)(26,33)(27,36)(29,32)(30,39)(31,37)(34,38)(35,40);(1,5,6,2)(3,7,4,8)(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,20,19,24)(25,29)(26,34)(27,37)(28,32)(30,40)(31,36)(33,38)(35,39)

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=ghi,e^2=i,f^2=gh,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, 8922696);