#433303

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=e,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,d^b=de,e^b=ei,h^b=hi,f^c=fi

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, 340499);