#358305

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,e^b=ei,h^b=hi,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, 265501);