#263530

order	512
nilpotency class	3
isoclinism	isoclinic class of #263527
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

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

normal subgroups only

This group has 914 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)(10,15)(12,26)(13,16)(17,31)(18,34)(19,20)(21,33)(22,38)(23,24)(27,39)(28,30)(37,40);(3,8)(5,6)(9,10,16,20,14,19,13,15)(11,21,24,17,25,31,23,33)(12,18,30,36,28,34,26,29)(22,37,39,32,27,40,38,35);(2,5)(3,8)(6,7)(9,11)(10,17)(12,27)(13,23)(14,25)(15,21)(16,24)(18,35)(19,33)(20,31)(22,28)(26,39)(29,40)(30,38)(32,34)(36,37);(9,12,16,30,14,28,13,26)(10,18,20,36,19,34,15,29)(11,22,24,39,25,27,23,38)(17,32,31,40,33,35,21,37)

Polycyclic presentation:

2^9|b^2=eh,d^2=eh,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,d^c=dh,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, 170726);