#357580

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×#121
prime radical	C₂²×#575	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4378 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)(3,4)(5,6)(7,8)(9,10,14,20,15,19,13,18)(11,21,23,36,25,16,24,31)(12,17,29,34,28,35,27,33)(22,37,26,32,38,30,39,40);(2,4)(9,11)(10,16)(12,26)(13,23)(14,24)(15,25)(17,32)(18,31)(19,21)(20,36)(22,29)(27,38)(28,39)(30,33)(34,37)(35,40);(6,8)(9,12)(10,17)(11,22)(13,27)(14,29)(15,28)(16,30)(18,33)(19,35)(20,34)(21,37)(23,26)(24,39)(25,38)(31,40)(32,36);(9,13,15,14)(10,18,19,20)(11,23,25,24)(12,27,28,29)(16,31,21,36)(17,33,35,34)(22,26,38,39)(30,40,37,32)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264776);