#436316

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=e,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 343512);