#7626420

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	C₂×#121	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6212 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)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,27,25)(23,31,32,28)(24,35,38,29)(26,39,40,30)(33,41,36,43)(34,42,46,45)(37,44,50,48)(47,52,49,51);(5,6,9,12)(7,10,15,13)(8,11,16,18)(14,17,20,19)(22,25)(23,32)(29,35)(30,39)(33,36)(34,46)(44,48)(47,49);(1,3,4,2)(5,6)(7,10)(8,11)(9,12)(13,15)(14,17)(16,18)(19,20)(21,23,27,32)(22,28,25,31)(24,36,38,33)(26,34,40,46)(29,43,35,41)(30,42,39,45)(37,49,50,47)(44,52,48,51);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,24)(22,29)(23,33)(25,35)(26,37)(27,38)(28,41)(30,44)(31,43)(32,36)(34,47)(39,48)(40,50)(42,51)(45,52)(46,49);(21,25,27,22)(23,31,32,28)(24,35,38,29)(26,39,40,30)(33,43,36,41)(34,45,46,42)(37,48,50,44)(47,52,49,51);(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,26,27,40)(22,30,25,39)(23,34,32,46)(24,37,38,50)(28,42,31,45)(29,44,35,48)(33,47,36,49)(41,51,43,52)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,e^2=h,f^2=g,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, 7533616);