#499625

order	512
nilpotency class	4
isoclinism	isoclinic class of #13371
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#476
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

495 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=eh,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,e^b=ei,f^b=fi,g^b=gh,h^b=hi,e^c=ei,g^c=ghi,h^c=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, 406821);