#8922622

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

4412 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)(3,7)(4,6)(5,8)(9,10,14,13)(11,15,17,19)(12,16,20,21)(18,22,23,24)(25,26)(27,34)(28,33)(29,32)(30,31)(35,40)(36,39)(37,38);(1,2,4,6)(3,5,8,7)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(27,29)(32,34)(35,36)(39,40);(1,2)(3,5)(4,6)(7,8)(11,17)(12,20)(15,19)(16,21)(25,27,31,29)(26,32,30,34)(28,36,38,35)(33,40,37,39);(1,3)(2,5)(4,8)(6,7)(9,12,14,20)(10,16,13,21)(11,18,17,23)(15,22,19,24)(25,28,31,38)(26,33,30,37)(27,35,29,36)(32,39,34,40);(9,13,14,10)(11,19,17,15)(12,16,20,21)(18,22,23,24)(25,29,31,27)(26,32,30,34)(28,35,38,36)(33,40,37,39);(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24)(25,30,31,26)(27,32,29,34)(28,37,38,33)(35,39,36,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,d^2=gi,e^2=gi,f^2=h,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, 8829818);