#8651461

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	#2541	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4295 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=gh,e^2=hi,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, 8558657);