#220398

order	512
nilpotency class	3
isoclinism	isoclinic class of #220264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1055 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=de,e^b=eh,g^c=gi,e^d=eh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 127594);