#10361092

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

4449 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)(25,26,30,32)(27,35,37,31)(28,33,38,36)(29,39,40,34);(1,2)(3,7)(4,6)(5,8)(10,13)(11,18)(16,22)(19,21)(25,27,30,37)(26,31,32,35)(28,34,38,39)(29,33,40,36);(1,2,4,6)(3,5,8,7)(12,20)(16,22)(19,21)(23,24)(26,32)(28,38)(31,35)(34,39);(1,2,4,6)(3,5,8,7)(9,11,14,18)(10,15,13,17)(12,21,20,19)(16,24,22,23)(25,27)(26,31)(28,34)(29,36)(30,37)(32,35)(33,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,19)(13,22)(14,20)(15,23)(17,24)(18,21)(25,28)(26,33)(27,34)(29,35)(30,38)(31,40)(32,36)(37,39);(1,3)(2,5)(4,8)(6,7)(9,13,14,10)(11,17,18,15)(12,22,20,16)(19,24,21,23)(25,29,30,40)(26,34,32,39)(27,36,37,33)(28,35,38,31)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=ghi,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, 10268288);