#237206

order	512
nilpotency class	3
isoclinism	isoclinic class of #10176
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#392
prime radical	#60415	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1409 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,7)(3,5,8,6)(9,10)(11,16)(12,24)(13,18)(14,19)(15,20)(17,28)(21,34)(22,31)(23,33)(25,30)(26,35)(27,36)(29,32)(37,39)(38,40);(2,5)(3,8)(6,7)(9,11)(10,16)(12,25)(13,22)(14,29)(15,23)(17,34)(18,31)(19,32)(20,33)(21,26)(24,39)(27,37)(28,38)(30,36)(35,40);(9,12)(10,17)(11,21)(13,26)(14,28)(15,27)(16,30)(18,24)(19,36)(20,35)(22,37)(23,38)(25,29)(31,34)(32,40)(33,39);(9,13,15,14)(10,18,20,19)(11,22,23,29)(12,26,27,28)(16,31,33,32)(17,24,35,36)(21,37,38,25)(30,34,39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 144402);