#216071

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
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

871 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)(11,18)(12,20,22,23)(13,17)(14,15)(16,19,24,21)(25,26,29,32,30,34,28,33)(27,31,36,38,37,40,35,39);(1,2)(3,6)(4,5)(7,8)(9,11,13,14)(10,15,17,18)(12,21,22,19)(16,20,24,23)(25,27,30,37)(26,31,34,40)(28,35,29,36)(32,38,33,39);(2,5)(6,8)(9,12)(10,16)(11,19)(13,22)(14,21)(15,23)(17,24)(18,20)(26,32)(28,29)(31,38)(33,34)(35,36)(39,40);(1,3,4,7)(2,6,5,8)(25,28,30,29)(26,32,34,33)(27,35,37,36)(31,38,40,39)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=ce,d^c=di,f^c=fh,g^c=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 123267);