#6532551

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500060
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

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

2874 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)(7,13)(10,14)(15,16)(18,19)(21,22,27,42,26,31,25,30)(23,32,41,28,34,48,35,44)(24,36,39,29,38,47,40,46)(33,43,50,49,37,45,51,52);(2,4)(5,6)(7,14)(8,11)(9,12)(10,13)(15,19)(16,18)(17,20)(21,23)(22,28)(24,37)(25,41)(26,34)(27,35)(29,45)(30,48)(31,44)(32,42)(33,38)(36,52)(39,50)(40,51)(43,46)(47,49);(5,7,9,13)(6,10,12,14)(8,16,17,15)(11,19,20,18)(21,24,26,38)(22,29,31,46)(23,33,34,37)(25,39,27,40)(28,43,44,45)(30,47,42,36)(32,49,48,52)(35,50,41,51);(5,8,9,17)(6,11,12,20)(7,15,13,16)(10,18,14,19)(23,34)(24,38)(28,44)(29,46)(32,48)(35,41)(36,47)(39,40);(6,12)(10,14)(11,20)(18,19)(21,25,26,27)(22,30,31,42)(23,35,34,41)(24,39,38,40)(28,32,44,48)(29,47,46,36)(33,50,37,51)(43,49,45,52)

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6439747);