#8696734

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	#51937	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4584 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,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, 8603930);