#8425564

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4394 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=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, 8332760);