#451234

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

782 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)(9,10,14,18)(11,20)(12,25)(13,16,29,19)(15,24)(17,26)(21,36,38,40)(22,32)(23,30)(27,35)(28,33)(31,39,34,37);(1,3)(2,5)(4,7)(6,8)(9,11,13,22,14,24,29,23)(10,15,16,30,18,20,19,32)(12,21,26,37,28,38,27,39)(17,34,33,36,35,31,25,40);(3,7)(5,8)(10,16,18,19)(15,30,20,32)(17,33,35,25)(31,40,34,36);(9,12)(10,17)(11,21)(13,26)(14,28)(15,31)(16,33)(18,35)(19,25)(20,34)(22,37)(23,39)(24,38)(27,29)(30,40)(32,36)

Polycyclic presentation:

2^9|a^2=efgh,b^2=ef,c^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,h^b=hi,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, 358430);