#216159

order	512
nilpotency class	3
isoclinism	isoclinic class of #32706
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂²×C₂²⋊C₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

859 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=e,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dehi,c^b=cei,d^c=di,f^c=fh,g^c=gh

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, 123355);