#433227

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

903 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)(5,6,11,14)(7,10,18,15)(8,12)(9,17,25,31)(13,29,23,24)(16,22)(19,30)(20,27)(21,28)(26,32)(33,36)(34,35);(5,7,15,14,11,18,10,6)(8,12,22,16,21,28,20,27)(9,23,29,31,25,13,24,17)(19,30,34,35,33,36,32,26);(5,8)(6,12)(7,16)(9,19)(10,20)(11,21)(13,26)(14,28)(15,22)(17,30)(18,27)(23,35)(24,32)(25,33)(29,34)(31,36);(2,4)(5,9,10,24,11,25,15,29)(6,13,7,17,14,23,18,31)(8,19,20,32,21,33,22,34)(12,26,16,30,28,35,27,36)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,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, 340423);