#10422713

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

4259 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=h,e^2=gh,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, 10329909);