#8436591

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	#52471	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4459 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)(3,7)(4,5)(6,8)(9,10,16,15)(11,20,22,17)(12,18,13,23)(14,24,21,19)(25,26,29,30)(27,33,35,31)(28,32,37,36)(34,39,40,38);(1,3)(2,7)(4,8)(5,6)(9,11,16,22)(10,17,15,20)(12,21,13,14)(18,24,23,19)(26,30)(31,33)(32,36)(38,39);(1,4)(2,3)(5,8)(6,7)(9,12,16,13)(10,18,15,23)(11,21,22,14)(17,24,20,19)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(1,5,6,3)(2,4,8,7)(9,13,16,12)(10,18,15,23)(11,14,22,21)(17,24,20,19)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,39)(38,40);(9,14)(10,19)(11,12)(13,22)(15,24)(16,21)(17,18)(20,23);(1,4)(2,5)(3,8)(6,7)(9,15,16,10)(11,20,22,17)(12,23,13,18)(14,24,21,19)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=ghi,d^2=gh,f^2=g,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, 8343787);