#281318

order	512
nilpotency class	3
isoclinism	isoclinic class of #264363
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

565 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)(3,5)(4,6)(7,8)(9,10,14,16)(11,19,20,17)(12,15,21,24)(13,22,23,18)(25,26)(27,31)(28,30)(29,32)(33,40)(34,37)(35,38)(36,39);(1,3,4,7)(2,5,6,8)(9,11,13,12,14,20,23,21)(10,15,18,19,16,24,22,17)(26,30)(27,33)(29,35)(31,39)(34,36)(37,40);(2,6)(5,8)(10,16)(15,24)(17,19)(18,22)(25,27,28,34)(26,31,30,37)(29,36,35,33)(32,39,38,40);(9,12,13,20,14,21,23,11)(10,17,18,15,16,19,22,24)(26,30)(31,37)(32,38)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 188514);