#8695954

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4428 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=g,e^2=h,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, 8603150);