#8930511

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

4187 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8837707);