#10108611

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

4647 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=i,d^2=h,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10015807);