#6393856

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×#3536	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3393 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)(11,17)(12,15,20,22)(13,16)(14,19)(18,23,24,21);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23);(9,12)(10,15,16,22)(11,18)(13,20)(14,21,17,23)(19,24);(2,5)(6,8)(11,19)(14,17)(18,24)(21,23);(1,3,4,7)(2,6,5,8)(10,16)(11,19)(15,22)(18,24)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,e^2=fi,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, 6301052);