#606830

order	512
nilpotency class	2
isoclinism	isoclinic class of #53158
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

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

1578 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fgh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dfh,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 514026);