#8471156

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

4619 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)(4,6)(5,8)(9,10,16,15)(11,19,14,17)(12,18,13,22)(20,24,21,23)(25,26,32,36)(27,33,30,31)(28,38,37,34)(29,35,39,40);(1,2,4,6)(3,5,8,7)(9,11,16,14)(10,17,15,19)(12,20,13,21)(18,23,22,24)(25,27,32,30)(26,33,36,31)(28,39,37,29)(34,35,38,40);(1,2)(3,5)(4,6)(7,8)(9,12,16,13)(10,18,15,22)(11,20,14,21)(17,23,19,24)(25,28)(26,34)(27,29)(30,39)(31,35)(32,37)(33,40)(36,38);(1,3)(2,5)(4,8)(6,7)(9,13,16,12)(10,18,15,22)(11,21,14,20)(17,23,19,24)(25,29,32,39)(26,35,36,40)(27,37,30,28)(31,38,33,34);(9,14,16,11)(10,19,15,17)(12,21,13,20)(18,24,22,23)(25,30,32,27)(26,31,36,33)(28,39,37,29)(34,35,38,40);(9,15,16,10)(11,19,14,17)(12,22,13,18)(20,24,21,23)(25,31)(26,27)(28,35)(29,34)(30,36)(32,33)(37,40)(38,39)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=ghi,d^2=gi,e^2=gi,f^2=ghi,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, 8378352);