#6370642

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

1916 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=f,e^2=fg,h^2=i,b^a=bf,d^a=df,e^a=ei,c^b=cfi,d^c=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, 6277838);