#8463787

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	#36396	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4361 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=gh,d^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,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, 8370983);