#10023270

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

4495 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=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, 9930466);