#453944

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

710 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 710 conjugacy classes of subgroups — too many to draw. Only its 162 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:

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

Polycyclic presentation:

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

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, 361140);