#310119

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ei,c^2=hi,d^2=fi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=eh,g^b=gi,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 217315);