#357812

order	512
nilpotency class	3
isoclinism	isoclinic class of #2018
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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

2690 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)(9,10,14,19,15,20,13,18)(11,21,24,33,25,32,23,16)(12,26,28,36,30,17,29,35)(22,34,38,40,39,37,27,31);(1,2,3,4)(9,11,13,23,15,25,14,24)(10,16,19,21,20,33,18,32)(12,27,29,39,30,38,28,22)(17,34,35,40,26,37,36,31);(6,8)(9,12)(10,17)(11,22)(13,28)(14,29)(15,30)(16,31)(18,35)(19,36)(20,26)(21,37)(23,38)(24,27)(25,39)(32,34)(33,40);(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,28,30,29)(16,32,33,21)(17,35,26,36)(22,38,39,27)(31,34,40,37)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=i,e^2=i,g^2=i,h^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,h^c=hi

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, 265008);