#8651420

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4225 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=gh,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8558616);