#358243

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1654 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)(5,6,8,7)(9,10)(11,21)(12,25)(13,18)(14,19)(15,20)(16,23)(17,27)(22,37)(24,32)(26,40)(28,35)(29,33)(30,36)(31,39)(34,38);(5,7,8,6)(9,11,13,23,15,24,14,29)(10,16,19,21,20,33,18,32)(12,26,27,22,28,38,30,39)(17,34,25,37,36,40,35,31);(6,7)(9,12,13,27,15,28,14,30)(10,17,19,25,20,36,18,35)(11,22,23,38,24,39,29,26)(16,31,21,34,33,37,32,40);(9,13,15,14)(10,18,20,19)(11,23,24,29)(12,27,28,30)(16,32,33,21)(17,35,36,25)(22,38,39,26)(31,40,37,34)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,g^b=gi,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, 265439);