#8505948

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4637 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8413144);