#352251

order	512
nilpotency class	3
isoclinism	isoclinic class of #352238
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2856
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#33190	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

796 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 796 conjugacy classes of subgroups — too many to draw. Only its 198 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259447);