#6524791

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500076
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3561	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2910 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6431987);