#187618

order	512
nilpotency class	3
isoclinism	isoclinic class of #18992
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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

2006 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 2006 conjugacy classes of subgroups — too many to draw. Only its 322 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:

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

Polycyclic presentation:

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

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, 94814);