#8922534

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

4554 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)(3,7)(4,5)(6,8)(9,10,14,13)(11,15,17,19)(12,16,20,21)(18,22,23,24)(26,31)(27,33)(29,37)(36,40);(2,5)(6,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,26,30,31)(27,32,33,28)(29,34,37,39)(35,40,38,36);(1,2)(3,6)(4,5)(7,8)(11,17)(12,20)(15,19)(16,21)(25,27,30,33)(26,32,31,28)(29,38,37,35)(34,36,39,40);(1,3,4,8)(2,6,5,7)(9,12,14,20)(10,16,13,21)(11,18,17,23)(15,22,19,24)(25,27)(26,32)(28,31)(29,35)(30,33)(34,40)(36,39)(37,38);(9,13,14,10)(11,19,17,15)(12,16,20,21)(18,22,23,24)(25,28,30,32)(26,33,31,27)(29,36,37,40)(34,38,39,35);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,29)(26,34)(27,35)(28,36)(30,37)(31,39)(32,40)(33,38)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=i,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8829730);