#10349024

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

4645 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)(3,7)(4,5)(6,8)(9,10)(11,20)(12,16)(13,17)(14,21)(15,19)(18,24)(22,23)(25,26,30,33)(27,34,28,31)(29,37,38,32)(35,39,36,40);(2,5)(6,7)(11,12)(14,24)(16,20)(18,21)(25,27,30,28)(26,31,33,34)(29,35,38,36)(32,39,37,40);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,19,20)(13,18,23,21)(14,22,24,17)(25,26)(27,34)(28,31)(29,32)(30,33)(35,40)(36,39)(37,38);(1,3,4,8)(2,6,5,7)(9,12,15,11)(10,16,19,20)(13,18,23,21)(14,17,24,22)(25,26)(27,31)(28,34)(29,32)(30,33)(35,39)(36,40)(37,38);(9,13,15,23)(10,17,19,22)(11,21,12,18)(14,16,24,20)(25,28,30,27)(26,31,33,34)(29,36,38,35)(32,39,37,40);(9,14,15,24)(10,18,19,21)(11,22,12,17)(13,16,23,20)(25,29,30,38)(26,32,33,37)(27,35,28,36)(31,39,34,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gi,d^2=gh,e^2=h,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, 10256220);