#8962941

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

4493 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)(3,5)(4,7)(6,8)(9,10)(11,12)(13,17)(14,22)(15,16)(18,23)(19,24)(20,21)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(1,2,6,8)(3,5,7,4)(9,11)(10,12)(13,15)(14,19)(16,17)(18,20)(21,23)(22,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,3,6,7)(2,4,8,5)(10,16)(11,13)(14,23)(20,24)(27,33)(29,37)(31,34)(32,36);(1,4,6,5)(2,3,8,7)(9,12)(10,11)(13,16)(14,20)(15,17)(18,19)(21,22)(23,24)(25,28,30,26)(27,34,33,31)(29,32,37,36)(35,38,39,40);(1,2)(3,4)(5,7)(6,8)(9,13,15,11)(10,17,16,12)(14,21,23,19)(18,24,22,20)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40);(1,5,6,4)(2,7,8,3)(9,14,15,23)(10,18,16,22)(11,19,13,21)(12,20,17,24)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,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, 8870137);