#299268

order	512
nilpotency class	3
isoclinism	isoclinic class of #299253
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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 927 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ehi,c^2=f,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,c^b=cg,g^b=gh,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206464);