#6370663

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1906 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=f,e^2=fg,g^2=i,b^a=bf,c^a=ci,d^a=df,e^a=ei,c^b=cfi,d^b=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277859);