#464738

order	512
nilpotency class	3
isoclinism	isoclinic class of #454025
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2002	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1815 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

(6,9)(10,12)(13,14,18,22,17,20,19,27)(15,23,24,28,16,25,26,21);(1,2,3,4)(5,6,8,9)(7,10,11,12)(14,20)(21,28)(22,27)(23,25);(2,4)(7,11)(10,12)(13,15)(14,21)(16,17)(18,26)(19,24)(20,28)(22,25)(23,27);(5,7,8,11)(6,10,9,12)(13,16,17,15)(14,21,20,28)(18,26,19,24)(22,23,27,25)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371934);