#432837

order	512
nilpotency class	3
isoclinism	isoclinic class of #417902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2679
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1565 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1565 conjugacy classes of subgroups — too many to draw. Only its 190 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72228050
Fitting subgroup of	#72228047
maximal subgroup of	#72209625, #72228035, #72228038, …
maximal normal subgroup of	#72209625, #72228035, #72228038, …
maximal quotient of	#72209694, C₂×#432837

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340033);