#352995

order	512
nilpotency class	3
isoclinism	isoclinic class of #352174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

874 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 874 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 52
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 52 points:

(2,4)(5,6)(7,13)(8,16)(9,11)(10,15)(12,17)(14,19)(18,20)(21,22,26,29)(23,31,35,45)(24,28,36,43)(25,38,32,30)(27,39,41,34)(33,44,47,52)(37,49,46,42)(40,50,51,48);(7,14)(10,18)(13,19)(15,20)(21,23,32,39,26,35,25,34)(22,27,38,45,29,41,30,31)(24,33,46,50,36,47,37,48)(28,40,49,52,43,51,42,44);(5,7,8,15,9,14,17,20)(6,10,12,13,11,18,16,19)(21,24)(22,28)(23,33)(25,37)(26,36)(27,40)(29,43)(30,42)(31,44)(32,46)(34,48)(35,47)(38,49)(39,50)(41,51)(45,52);(1,2,3,4)(6,11)(10,18)(12,16)(13,19)(22,29)(24,36)(27,41)(30,38)(31,45)(33,47)(37,46)(48,50)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=f,d^2=ghi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,c^b=ci,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260191);