#10325395

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

4350 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)(9,10,14,18)(11,15,19,24)(12,16,22,21)(13,17,23,20)(25,26,32,35)(27,33,37,39)(28,38,29,34)(30,31,40,36);(2,7)(3,5)(11,19)(13,23)(15,24)(17,20)(25,27,32,37)(26,33,35,39)(28,36,29,31)(30,34,40,38);(2,7)(4,8)(9,11)(10,15)(12,17)(13,21)(14,19)(16,23)(18,24)(20,22)(25,28,32,29)(26,34,35,38)(27,36,37,31)(30,33,40,39);(1,3,6,5)(2,4,7,8)(9,11)(10,15)(12,17)(13,21)(14,19)(16,23)(18,24)(20,22)(25,29,32,28)(26,34,35,38)(27,31,37,36)(30,33,40,39);(1,4,6,8)(2,3,7,5)(9,12)(10,16)(11,20)(13,15)(14,22)(17,19)(18,21)(23,24)(25,30)(26,31)(27,34)(28,33)(29,39)(32,40)(35,36)(37,38);(1,5,6,3)(2,8,7,4)(9,13)(10,17)(11,21)(12,15)(14,23)(16,19)(18,20)(22,24)(25,31)(26,30)(27,28)(29,37)(32,36)(33,34)(35,40)(38,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=h,e^2=g,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, 10232591);