#263435

order	512
nilpotency class	3
isoclinism	isoclinic class of #263431
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
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

858 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 858 conjugacy classes of subgroups — too many to draw. Only its 144 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)(9,10,15,20)(11,17,23,32)(12,24,28,39)(13,29,14,19)(16,30,33,22)(18,27,36,26)(21,37,25,31)(34,40,35,38);(3,8)(5,6)(9,11,14,30,15,23,13,22)(10,16,29,32,20,33,19,17)(12,21,27,40,28,25,26,38)(18,31,39,35,36,37,24,34);(2,5)(3,8)(6,7)(9,11)(10,17)(12,25)(13,22)(14,30)(15,23)(16,29)(18,35)(19,33)(20,32)(21,28)(24,31)(26,40)(27,38)(34,36)(37,39);(9,12,14,27,15,28,13,26)(10,18,29,39,20,36,19,24)(11,21,30,40,23,25,22,38)(16,31,32,35,33,37,17,34)

Polycyclic presentation:

2^9|a^2=h,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=gi,c^b=cg,f^b=fi,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, 170631);