#126825

order	512
nilpotency class	3
isoclinism	isoclinic class of #1504
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=g,d^2=g,e^2=h,f^2=gh,h^2=i,b^a=bd,d^a=dg,f^a=fgi,c^b=cgi,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34021);