#470564

order	512
nilpotency class	3
isoclinism	isoclinic class of #459487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2782
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

596 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)(5,6,10,15,11,16,9,12)(7,17,19,13,20,27,25,29)(8,14,24,30,21,32,23,26)(18,33,22,36,34,31,35,28);(6,12,16,15)(8,21)(13,27,29,17)(14,30,32,26)(18,34)(22,35)(23,24)(28,31,36,33);(5,7,10,19,11,20,9,25)(6,13,12,17,16,29,15,27)(8,22,24,34,21,35,23,18)(14,31,26,36,32,33,30,28);(2,4)(5,8,9,23,11,21,10,24)(6,14,12,26,16,32,15,30)(7,18,19,22,20,34,25,35)(13,28,27,33,29,36,17,31)

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=f,d^2=eg,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=di,d^c=de,e^c=ei,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377760);