#8893459

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

4636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8800655);