#451807

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
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	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

803 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)(11,19)(12,24,28,17)(13,30)(14,31)(15,18)(16,22)(20,38,25,36)(21,33)(23,37)(26,29,35,27)(32,39,34,40);(1,3)(2,5)(4,7)(6,8)(9,11,14,22,15,23,30,33)(10,16,13,19,18,21,31,37)(12,25,27,40,28,20,29,39)(17,34,26,38,24,32,35,36);(3,7)(5,8)(9,12)(10,17)(11,20)(13,26)(14,27)(15,28)(16,32)(18,24)(19,36)(21,34)(22,39)(23,25)(29,30)(31,35)(33,40)(37,38);(9,13,14,10,15,31,30,18)(11,21,22,19,23,16,33,37)(12,26,27,17,28,35,29,24)(20,34,39,36,25,32,40,38)

Polycyclic presentation:

2^9|a^2=egh,b^2=e,d^2=ef,e^2=i,h^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=dei,h^c=hi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359003);