#263500

order	512
nilpotency class	3
isoclinism	isoclinic class of #263499
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

872 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 872 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=eh,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,f^b=fhi,g^b=gh,f^c=fi,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170696);