#8199303

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4301 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:

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

Polycyclic presentation:

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

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, 8106499);