#6672175

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1709 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)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22,25,29)(23,31,33,27)(24,35,37,28)(26,30,39,45)(32,46,40,41)(34,42,49,47)(36,44,38,43)(48,51,50,52);(5,7,9,15)(6,10,12,13)(8,14,17,20)(11,18,16,19)(23,32)(24,36)(26,39)(27,41)(28,43)(30,45)(31,46)(33,40)(34,49)(35,44)(37,38)(42,47);(2,4)(21,23,25,33)(22,27,29,31)(24,34,37,49)(26,40,39,32)(28,42,35,47)(30,46,45,41)(36,50,38,48)(43,52,44,51);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(23,33)(27,31)(32,40)(34,49)(41,46)(42,47)(48,50)(51,52);(7,15)(10,13)(14,20)(18,19)(21,24,25,37)(22,28,29,35)(23,34,33,49)(26,36,39,38)(27,42,31,47)(30,43,45,44)(32,48,40,50)(41,51,46,52)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6579371);