#212680

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
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 ◁ 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=e,d^2=gh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,c^b=chi,d^b=di,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, 119876);