#10422651

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

4668 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,d^2=gh,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10329847);