#342023

order	512
nilpotency class	3
isoclinism	isoclinic class of #333271
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

649 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)(9,10,12,16,13,17,22,15)(11,18,19,23,20,24,21,14);(1,3)(2,5)(4,8)(6,7)(25,26,29,32)(27,33,35,40)(28,36,38,31)(30,37,39,34);(9,11,13,20)(10,14,17,23)(12,21,22,19)(15,18,16,24)(25,27,29,35)(26,30,32,39)(28,37,38,34)(31,40,36,33);(2,6)(3,8)(10,15)(12,22)(14,18)(16,17)(19,21)(23,24)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 249219);