#194586

order	512
nilpotency class	3
isoclinism	isoclinic class of #19086
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	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1090 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101782);