#8824164

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

4414 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gi,d^2=hi,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8731360);