#497483

order	512
nilpotency class	4
isoclinism	isoclinic class of #480860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#547 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1059 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=ei,d^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=de,g^c=gh,g^f=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404679);