#8497874

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

4660 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=g,d^2=g,e^2=g,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, 8405070);