#232107

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

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

535 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ef,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=dei,d^c=dhi

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, 139303);