#6370945

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342574
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4469 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=h,d^2=f,e^2=i,b^a=bf,c^a=cg,g^a=gi,e^b=ei,h^c=hi,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278141);