#7626562

order	512
nilpotency class	2
isoclinism	isoclinic class of #62862
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6036 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=h,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh,d^c=dh

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533758);