#231348

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
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, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

538 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138544);