#451806

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
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

789 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)(12,23,27,17)(13,29)(14,30)(15,18)(16,21)(20,37,24,35)(22,36)(25,28,34,26)(31,38,33,39)(32,40);(1,3)(2,5)(4,7)(6,8)(9,11,14,21,15,22,29,40)(10,16,13,19,18,32,30,36)(12,24,26,39,27,20,28,38)(17,33,25,37,23,31,34,35);(3,7)(5,8)(9,12)(10,17)(11,20)(13,25)(14,26)(15,27)(16,31)(18,23)(19,35)(21,38)(22,24)(28,29)(30,34)(32,33)(36,37)(39,40);(9,13,14,10,15,30,29,18)(11,16,21,36,22,32,40,19)(12,25,26,17,27,34,28,23)(20,31,38,37,24,33,39,35)

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=de,h^c=hi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359002);