#105804

order	512
nilpotency class	3
isoclinism	isoclinic class of #105782
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

971 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,d^b=di,f^c=fi

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, 13000);