#8926622

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

4185 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)(9,10,16,21)(11,22,24,17)(12,18,14,20)(13,19,23,15)(27,33)(31,34)(35,39)(38,40);(1,2,6,8)(3,5,7,4)(9,11,16,24)(10,17,21,22)(12,23,14,13)(15,18,19,20)(25,26,30,28)(27,31,33,34)(29,36,37,32)(35,40,39,38);(1,3,6,7)(2,4,8,5)(9,12,16,14)(10,18,21,20)(11,23,24,13)(15,17,19,22)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,4,6,5)(2,3,8,7)(9,13)(10,19)(11,12)(14,24)(15,21)(16,23)(17,20)(18,22)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38);(1,2)(3,4)(5,7)(6,8)(9,14,16,12)(10,20,21,18)(11,23,24,13)(15,17,19,22)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40);(1,5,6,4)(2,7,8,3)(9,15,16,19)(10,13,21,23)(11,18,24,20)(12,22,14,17)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833818);