#358263

order	512
nilpotency class	3
isoclinism	isoclinic class of #23400
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1822 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,e^b=ei,g^b=gi,h^b=hi,f^c=fi,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, 265459);