#8917896

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

4449 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=g,e^2=i,f^2=g,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, 8825092);