#8746078

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

4312 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=i,d^2=hi,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8653274);