#9625485

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

4334 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)(11,13)(12,16)(14,15)(17,21)(18,20)(19,22)(23,24)(25,26,31,29)(27,28,35,32)(30,37,39,33)(34,38,40,36);(1,2)(3,4)(5,6)(7,8)(12,19)(16,22)(18,23)(20,24)(25,27,31,35)(26,32,29,28)(30,34,39,40)(33,38,37,36);(2,6)(3,7)(10,15)(11,17)(16,22)(18,23)(26,29)(28,32)(33,37)(36,38);(3,7)(4,8)(9,11,14,17)(10,13,15,21)(12,18,19,23)(16,20,22,24)(25,28)(26,27)(29,35)(30,38)(31,32)(33,40)(34,37)(36,39);(1,3)(2,4)(5,7)(6,8)(9,12,14,19)(10,16,15,22)(11,18,17,23)(13,20,21,24)(25,29,31,26)(27,32,35,28)(30,37,39,33)(34,38,40,36);(1,4)(2,3)(5,8)(6,7)(9,13)(10,17)(11,15)(12,20)(14,21)(16,23)(18,22)(19,24)(25,30)(26,33)(27,34)(28,36)(29,37)(31,39)(32,38)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9532681);