#219137

order	512
nilpotency class	3
isoclinism	isoclinic class of #29592
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	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

564 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)(11,17)(12,14)(13,21)(15,19)(16,20)(18,23)(22,24)(25,26,31,27,30,34,29,33)(28,32,37,35,38,40,39,36);(1,2)(3,6)(4,5)(7,8)(10,14)(11,18)(16,21)(19,24)(25,27,31,34,30,33,29,26)(28,36,37,32,38,35,39,40);(2,5)(6,8)(9,11,13,19,12,18,20,24)(10,15,16,17,14,22,21,23)(25,28)(26,32)(27,35)(29,39)(30,38)(31,37)(33,36)(34,40);(1,3,4,7)(2,6,5,8)(10,14)(15,22)(16,21)(17,23)(25,29,30,31)(26,33,34,27)(28,37,38,39)(32,35,40,36)

Polycyclic presentation:

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

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, 126333);