#217999

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

586 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)(9,10,15,19)(11,16,12,21)(13,18,14,23)(17,20,24,22)(25,26,31,33,30,28,29,34)(27,35,36,39,37,32,38,40);(2,5)(3,7)(9,11,15,12)(10,16,19,21)(13,22,14,20)(17,23,24,18)(25,27,31,36,30,37,29,38)(26,32,33,40,28,35,34,39);(1,3,4,7)(2,6,5,8)(9,12,15,11)(10,17,19,24)(13,22,14,20)(16,23,21,18)(25,28)(26,30)(27,35)(29,33)(31,34)(32,37)(36,39)(38,40);(2,5)(6,8)(9,13,15,14)(10,18,19,23)(11,20,12,22)(16,24,21,17)(25,29,30,31)(26,33,28,34)(27,36,37,38)(32,39,35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=eh,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cei,d^b=dh,d^c=di,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 125195);