#8718950

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	#53585	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4314 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=i,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8626146);