#7627640

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

6056 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)(22,23)(24,34)(26,39)(29,32)(35,46)(37,49)(41,44)(50,51);(7,14)(8,16)(11,13)(12,17)(21,22,27,23)(24,31,34,28)(25,32,38,29)(26,33,39,30)(35,40,46,43)(36,41,47,44)(37,42,49,45)(48,51,52,50);(6,10)(11,13)(12,17)(18,19)(21,23,27,22)(24,28,34,31)(25,29,38,32)(26,33,39,30)(35,43,46,40)(36,41,47,44)(37,42,49,45)(48,51,52,50);(5,7,9,14)(6,11,10,13)(8,15,16,20)(12,18,17,19)(21,24,27,34)(22,28,23,31)(25,35,38,46)(26,36,39,47)(29,40,32,43)(30,41,33,44)(37,48,49,52)(42,50,45,51);(5,6)(7,11)(8,12)(9,10)(13,14)(15,18)(16,17)(19,20)(21,25,27,38)(22,29,23,32)(24,35,34,46)(26,37,39,49)(28,40,31,43)(30,42,33,45)(36,48,47,52)(41,50,44,51);(5,8)(6,12)(7,15)(9,16)(10,17)(11,18)(13,19)(14,20)(21,26)(22,30)(23,33)(24,36)(25,37)(27,39)(28,41)(29,42)(31,44)(32,45)(34,47)(35,48)(38,49)(40,50)(43,51)(46,52)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=h,e^2=g,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, 7534836);