#470188

order	512
nilpotency class	3
isoclinism	isoclinic class of #454974
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

643 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6,10,15,11,16,9,14)(7,17,21,29,20,28,19,12)(8,22,25,31,26,13,24,32)(18,30,23,36,34,35,33,27);(5,7,9,19,11,20,10,21)(6,12,15,17,16,29,14,28)(8,23,25,34,26,33,24,18)(13,30,31,27,22,35,32,36);(5,8,9,24,11,26,10,25)(6,13,15,32,16,22,14,31)(7,18,21,23,20,34,19,33)(12,27,28,35,29,36,17,30);(2,4)(6,14,16,15)(8,24,26,25)(12,28,29,17)(18,33,34,23)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=ef,d^2=eg,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,f^c=fi,h^c=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, 377384);