#7627679

order	512
nilpotency class	2
isoclinism	isoclinic class of #62972
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×#532	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6310 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

134 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534875);