#470276

order	512
nilpotency class	3
isoclinism	isoclinic class of #459487
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

1197 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)(19,31)(20,30)(21,25)(22,24)(27,33)(29,35)(34,36);(6,11,15,14)(8,19)(12,26,28,24)(13,29,31,25)(16,33)(20,34)(21,35)(27,30,32,36);(2,4)(5,7)(6,12)(8,20)(9,22)(10,18)(11,24)(13,30)(14,26)(15,28)(16,21)(17,23)(19,34)(25,32)(27,29)(31,36)(33,35);(5,8,9,21,10,19,23,35)(6,13,11,25,15,31,14,29)(7,16,17,20,18,33,22,34)(12,27,26,36,28,32,24,30)

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^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, 377472);