#8746070

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

4303 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)(9,10,14,18)(11,13,17,15)(12,20,22,16)(19,21,24,23)(26,31)(28,34)(32,38)(36,40);(1,2)(3,7)(4,5)(6,8)(9,10,14,18)(11,13,17,15)(12,16,22,20)(19,23,24,21)(27,35)(32,38)(36,40)(37,39);(1,2)(3,7)(4,5)(6,8)(9,11,14,17)(10,15,18,13)(12,19,22,24)(16,21,20,23)(25,26)(27,32)(28,33)(29,34)(30,31)(35,38)(36,39)(37,40);(1,3,4,8)(2,6,5,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,21)(20,23)(22,24)(25,27,30,35)(26,32,31,38)(28,36,34,40)(29,37,33,39);(1,2)(3,6)(4,5)(7,8)(9,12,14,22)(10,16,18,20)(11,19,17,24)(13,23,15,21)(25,28,30,34)(26,33,31,29)(27,36,35,40)(32,39,38,37);(1,3)(2,7)(4,8)(5,6)(9,13,14,15)(10,17,18,11)(12,21,22,23)(16,19,20,24)(25,29,30,33)(26,34,31,28)(27,37,35,39)(32,40,38,36)

Polycyclic presentation:

2^9|a^2=i,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, 8653266);