#6687155

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	#42887	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1807 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,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, 6594351);