#299310

order	512
nilpotency class	3
isoclinism	isoclinic class of #299301
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3508	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

953 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 953 conjugacy classes of subgroups — too many to draw. Only its 174 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ei,c^2=f,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,c^b=cg,g^b=gh,d^c=dh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206506);