#8459161

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

4400 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:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=gh,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8366357);