#6675347

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667556
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1743 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)(5,6)(7,14)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22,26,24)(23,31,35,28)(25,36,38,29)(27,30,40,37)(32,34,44,41)(33,42,49,45)(39,50,51,43)(46,52,47,48);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,23,26,35)(22,28,24,31)(25,33,38,49)(27,41,40,34)(29,42,36,45)(30,44,37,32)(39,52,51,48)(43,47,50,46);(2,4)(21,24,26,22)(23,32,35,44)(25,29,38,36)(27,30,40,37)(28,41,31,34)(33,47,49,46)(39,50,51,43)(42,48,45,52);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25,26,38)(22,29,24,36)(23,33,35,49)(27,39,40,51)(28,42,31,45)(30,43,37,50)(32,46,44,47)(34,48,41,52);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(23,34)(27,40)(28,32)(30,37)(31,44)(33,48)(35,41)(39,51)(42,46)(43,50)(45,47)(49,52)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

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, 6582543);