#194605

order	512
nilpotency class	3
isoclinism	isoclinic class of #19050
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1090 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1090 conjugacy classes of subgroups — too many to draw. Only its 258 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,c^2=h,d^2=i,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,f^a=fh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fhi,g^c=gi

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, 101801);