#10424604

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

4739 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=i,e^2=g,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, 10331800);