#217682

order	512
nilpotency class	3
isoclinism	isoclinic class of #32706
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	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

672 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:

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

Polycyclic presentation:

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

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, 124878);