#8824407

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
prime radical	#46315	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4270 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,17)(14,15)(16,19)(18,21)(20,24)(22,23)(25,26,30,28)(27,33,35,31)(29,32,37,36)(34,39,40,38);(1,2,4,6)(3,5,8,7)(10,15)(13,16)(17,23)(21,24)(29,37)(32,36)(34,40)(38,39);(1,2)(3,5)(4,6)(7,8)(9,11,14,19)(10,13,15,16)(12,20,22,18)(17,24,23,21)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,18)(13,15)(14,19)(17,24)(20,22)(21,23)(25,28,30,26)(27,33,35,31)(29,36,37,32)(34,39,40,38);(9,12,14,22)(10,17,15,23)(11,18,19,20)(13,21,16,24)(25,29,30,37)(26,32,28,36)(27,34,35,40)(31,38,33,39);(9,13,14,16)(10,11,15,19)(12,21,22,24)(17,18,23,20)(25,27)(26,31)(28,33)(29,34)(30,35)(32,38)(36,39)(37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=hi,e^2=ghi,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, 8731603);