#358472

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

1958 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,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, 265668);