#10349028

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

4510 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)(3,4)(5,8)(6,7)(9,10,14,16)(11,17,19,13)(12,15,20,23)(18,24,21,22)(25,26)(27,36)(28,32)(29,33)(30,37)(31,35)(34,40)(38,39);(2,7)(3,5)(9,11,14,19)(10,13,16,17)(12,18,20,21)(15,22,23,24)(27,28)(30,40)(32,36)(34,37);(2,7)(4,8)(12,20)(15,23)(18,21)(22,24)(25,27,31,28)(26,32,35,36)(29,34,39,37)(30,38,40,33);(1,3,6,5)(2,8,7,4)(9,11)(10,13)(12,21)(14,19)(15,24)(16,17)(18,20)(22,23)(25,28,31,27)(26,32,35,36)(29,34,39,37)(30,33,40,38);(1,4)(2,3)(5,7)(6,8)(9,12,14,20)(10,15,16,23)(11,18,19,21)(13,22,17,24)(25,29,31,39)(26,33,35,38)(27,37,28,34)(30,32,40,36);(1,5,6,3)(2,4,7,8)(9,13)(10,11)(12,22)(14,17)(15,18)(16,19)(20,24)(21,23)(25,30,31,40)(26,34,35,37)(27,38,28,33)(29,32,39,36)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=i,d^2=h,e^2=gh,f^2=h,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, 10256224);