#363379

order	512
nilpotency class	3
isoclinism	isoclinic class of #190432
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1189 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

(9,10,12,16,13,14,19,22)(11,15,18,23,17,21,20,24);(1,2)(3,6)(4,5)(7,8)(10,14)(12,19)(15,21)(18,20);(2,5)(6,8)(9,11)(10,15,14,21)(12,20)(13,17)(16,24,22,23)(18,19);(1,3,4,7)(2,6,5,8)(10,14)(12,19)(15,21)(18,20)

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=g,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,h^b=hi,f^c=fi,h^c=hi,h^d=hi

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, 270575);