#8703529

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	#57612	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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,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, 8610725);