#445978

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	#2563
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

736 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=e,c^2=fg,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^d=ei,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, 353174);