#453909

order	512
nilpotency class	3
isoclinism	isoclinic class of #417902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2804
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

712 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 170 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

2^9|b^2=gh,c^2=fg,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,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, 361105);