#8422205

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	#42887	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4376 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8329401);