#10457913

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

4431 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)(3,7,8,6)(9,10,15,20)(11,16,22,24)(12,13,18,17)(14,23,21,19)(25,26,31,35)(27,30,29,32)(28,34,39,33)(36,40,37,38);(1,3)(2,6)(4,7)(5,8)(9,11,15,22)(10,16,20,24)(12,21,18,14)(13,19,17,23)(25,27,31,29)(26,32,35,30)(28,37,39,36)(33,38,34,40);(9,12)(10,17)(11,21)(13,20)(14,22)(15,18)(16,23)(19,24)(25,28)(26,33)(27,36)(29,37)(30,38)(31,39)(32,40)(34,35);(1,2,5,4)(3,7,8,6)(9,12)(10,13)(11,21)(14,22)(15,18)(16,19)(17,20)(23,24)(25,29,31,27)(26,30,35,32)(28,37,39,36)(33,38,34,40);(1,3)(2,7)(4,6)(5,8)(9,13)(10,18)(11,19)(12,20)(14,16)(15,17)(21,24)(22,23)(25,28)(26,34)(27,37)(29,36)(30,38)(31,39)(32,40)(33,35);(1,4,5,2)(3,7,8,6)(9,14,15,21)(10,19,20,23)(11,12,22,18)(13,16,17,24)(25,30)(26,27)(28,38)(29,35)(31,32)(33,36)(34,37)(39,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10365109);