#481501

order	512
nilpotency class	4
isoclinism	isoclinic class of #481477
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1208 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,e^2=hi,b^a=be,e^a=eg,g^a=gi,e^b=eh,f^b=fghi,d^c=di,f^e=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388697);