#453807

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	#2571
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

589 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)(11,14)(12,21,26,23)(13,22)(15,19)(16,20)(17,25,32,27)(18,29)(24,37,39,31)(28,34)(30,33)(35,38,36,40);(1,2,5,6)(3,7,4,8)(9,11,14,20,16,19,15,10)(12,23,25,17,26,21,27,32)(13,18,28,22,30,34,29,33)(24,37,38,36,39,31,40,35);(1,3,5,4)(2,7,6,8)(9,12,16,26)(10,17)(11,21)(13,24,30,39)(14,25,15,27)(18,31)(19,23)(20,32)(22,35)(28,38,29,40)(33,36)(34,37);(1,4,5,3)(2,7,6,8)(9,13,14,28,16,30,15,29)(10,18,19,33,20,34,11,22)(12,24,25,38,26,39,27,40)(17,31,23,36,32,37,21,35)

Polycyclic presentation:

2^9|a^2=g,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, 361003);