#8720342

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

4478 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gi,d^2=ghi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8627538);