#433301

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1060 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:

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

Polycyclic presentation:

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

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, 340497);