#499686

order	512
nilpotency class	4
isoclinism	isoclinic class of #499266
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

556 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 130 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=efhi,g^2=hi,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=dh,g^c=gh,h^c=hi,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239800, #72239816, #72239836, …
maximal normal subgroup of	#72239800, #72239816, #72239836, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406882);