#8932216

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

4187 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=gh,d^2=gh,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8839412);