#203983

order	512
nilpotency class	3
isoclinism	isoclinic class of #19230
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

719 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=f,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,e^b=eh,d^c=di

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111179);