#10340215

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

4353 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=g,e^2=h,f^2=g,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, 10247411);