#8926682

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

4798 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:

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

Polycyclic presentation:

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

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, 8833878);