#8688210

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

4536 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,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, 8595406);