#8664383

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	#36471	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4329 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8571579);