#220420

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×#3354	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

951 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,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, 127616);