#6529440

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×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2743 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 6436636);