#8900111

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

4360 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8807307);