#470275

order	512
nilpotency class	3
isoclinism	isoclinic class of #459486
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2684
prime radical	#23399	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1207 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)(7,12)(8,13)(9,11)(10,15)(14,23)(16,32)(17,26)(18,28)(19,29)(20,25)(21,31)(22,24)(27,33)(30,35)(34,36);(6,11,15,14)(12,26,28,24)(13,25,31,30)(27,36,32,29);(2,4)(5,7)(6,12)(8,19)(9,22)(10,18)(11,24)(13,29)(14,26)(15,28)(16,20)(17,23)(21,34)(25,32)(27,30)(31,36)(33,35);(5,8,9,20,10,21,23,35)(6,13,11,25,15,31,14,30)(7,16,17,19,18,33,22,34)(12,27,26,36,28,32,24,29)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377471);