#342104

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, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

626 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)(3,7)(4,6)(5,8)(9,10,15,24,14,19,11,16)(12,22,20,17,13,23,21,18);(1,3)(2,5)(4,8)(6,7)(9,11,14,15)(10,16,19,24)(12,20,13,21)(17,23,18,22)(25,26,29,32)(27,33,35,40)(28,36,38,31)(30,37,39,34);(9,12,14,13)(10,17,19,18)(11,20,15,21)(16,23,24,22)(25,27,29,35)(26,30,32,39)(28,37,38,34)(31,40,36,33);(2,6)(3,8)(9,13,14,12)(10,18,19,17)(11,21,15,20)(16,22,24,23)

Polycyclic presentation:

2^9|a^2=fh,b^2=hi,c^2=hi,d^2=h,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, 249300);