#8506266

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

4368 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,e^2=gi,f^2=hi,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, 8413462);