#8346167

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	#58087	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4326 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gi,d^2=gi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8253363);