#10322372

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

4690 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=gh,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, 10229568);