#8757106

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

4489 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=hi,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664302);