#6370654

order	512
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂⁴
commutator	C₂²	C₂³×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂⁴×C₄
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ 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)(21,22,25,30,26,28,36,41)(23,27,33,39,31,37,44,48)(24,29,34,42,35,40,46,50)(32,38,45,49,43,47,51,52);(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)(28,37)(29,38)(30,39)(34,45)(35,43)(36,44)(40,47)(41,48)(42,49)(46,51)(50,52);(7,15)(11,14)(16,17)(19,20)(22,28)(23,31)(29,40)(30,41)(32,43)(33,44)(42,50)(45,51);(5,8,10,9)(6,12,13,18)(7,16,15,17)(11,19,14,20)(23,31)(27,37)(32,43)(33,44)(38,47)(39,48)(45,51)(49,52);(5,9,10,8)(6,12,13,18)(7,17,15,16)(11,19,14,20)(21,24,25,34,26,35,36,46)(22,29,30,42,28,40,41,50)(23,32,33,45,31,43,44,51)(27,38,39,49,37,47,48,52)

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,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, 6277850);