#418785

order	512
nilpotency class	3
isoclinism	isoclinic class of #417460
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35659	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2190 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^c=fi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325981);