#6393853

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342866
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

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	C₂×#3558	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#124 ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3459 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6301049);