#453817

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

569 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=efh,c^2=ef,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,g^c=gi

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, 361013);