#8746027

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4183 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)(3,5)(4,7)(6,8)(9,10,15,17)(11,18,13,16)(12,14,19,24)(20,21,22,23)(26,28)(31,34)(32,36)(38,40);(2,3)(6,7)(9,11,15,13)(10,16,17,18)(12,22,19,20)(14,21,24,23)(27,33)(31,34)(35,39)(38,40);(1,2,5,3)(4,6,8,7)(10,17)(12,19)(16,18)(20,22)(25,26,30,28)(27,31,33,34)(29,32,37,36)(35,38,39,40);(1,3,5,2)(4,7,8,6)(9,12)(10,14)(11,20)(13,22)(15,19)(16,23)(17,24)(18,21)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(1,4,5,8)(2,6,3,7)(9,13,15,11)(10,18,17,16)(12,22,19,20)(14,21,24,23)(25,28,30,26)(27,34,33,31)(29,36,37,32)(35,40,39,38);(9,14)(10,19)(11,21)(12,17)(13,23)(15,24)(16,20)(18,22)(25,29,30,37)(26,32,28,36)(27,35,33,39)(31,38,34,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=ghi,d^2=i,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8653223);