#6741212

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667836
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2428	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1295 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=f,d^2=g,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6648408);