#6673404

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	C₂×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1741 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)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,27,25)(23,28,35,32)(24,29,38,39)(26,30,42,40)(31,43,49,50)(33,44,34,45)(36,46,51,52)(37,47,41,48);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23)(22,28)(24,36)(25,32)(26,33)(27,35)(29,46)(30,44)(31,37)(34,42)(38,51)(39,52)(40,45)(41,49)(43,47)(48,50);(2,4)(21,24,27,38)(22,29,25,39)(23,31,35,49)(26,41,42,37)(28,43,32,50)(30,48,40,47)(33,36,34,51)(44,46,45,52);(6,11)(10,13)(12,19)(18,20)(21,25,27,22)(23,32,35,28)(24,29,38,39)(26,40,42,30)(31,43,49,50)(33,45,34,44)(36,46,51,52)(37,47,41,48);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(23,33)(26,42)(28,44)(30,40)(31,36)(32,45)(34,35)(37,41)(43,46)(47,48)(49,51)(50,52)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=f,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, 6580600);