#449435

order	512
nilpotency class	3
isoclinism	isoclinic class of #417901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×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	#2684
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

704 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 182 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=g,d^2=eg,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^b=ei,f^b=fi,h^b=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356631);