#10397292

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

4449 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:

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

Polycyclic presentation:

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

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, 10304488);