#351763

order	512
nilpotency class	3
isoclinism	isoclinic class of #347668
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

662 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=fi,d^2=i,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,d^c=dh,g^d=gi

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, 258959);