#8734898

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	#59074	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4408 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)(4,6)(5,8)(9,10)(11,16)(12,15)(13,17)(14,19)(18,23)(20,21)(22,24)(25,26,31,32)(27,35,29,33)(28,34,39,38)(30,36,40,37);(1,2,4,6)(3,5,8,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,24)(21,23)(26,32)(28,39)(33,35)(36,37);(1,2)(3,5)(4,6)(7,8)(12,20)(15,21)(18,24)(22,23)(25,27,31,29)(26,33,32,35)(28,36,39,37)(30,38,40,34);(1,3,4,8)(2,5,6,7)(9,12,13,20)(10,15,17,21)(11,18,19,24)(14,22,16,23)(25,28,31,39)(26,34,32,38)(27,37,29,36)(30,33,40,35);(9,11)(10,16)(12,18)(13,19)(14,17)(15,23)(20,24)(21,22)(25,29,31,27)(26,35,32,33)(28,36,39,37)(30,38,40,34);(9,10)(11,14)(12,21)(13,17)(15,20)(16,19)(18,23)(22,24)(25,30)(26,36)(27,38)(28,35)(29,34)(31,40)(32,37)(33,39)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8642094);