#451756

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

906 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)(9,10,14,20,15,19,13,17)(11,21,23,32,24,16,29,30)(12,25,28,18,26,34,27,35)(22,36,38,33,37,39,40,31);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,22)(13,23)(14,29)(15,24)(17,30)(18,33)(19,21)(20,32)(25,39)(26,37)(27,38)(28,40)(31,35)(34,36);(3,7)(5,8)(10,17,19,20)(12,26)(16,30,21,32)(18,34,35,25)(22,37)(27,28)(31,39,33,36)(38,40);(9,12,13,27,15,26,14,28)(10,18,20,34,19,35,17,25)(11,22,23,38,24,37,29,40)(16,31,32,39,21,33,30,36)

Polycyclic presentation:

2^9|a^2=egh,c^2=e,d^2=f,f^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^b=hi,d^c=di

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, 358952);