#6672161

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1680 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);(5,7,9,15)(6,10,12,13)(8,14,17,20)(11,18,16,19)(21,22)(23,32)(24,29)(25,38)(26,40)(27,31)(28,35)(30,41)(33,43)(34,49)(36,44)(37,46)(39,45)(42,50)(47,51)(48,52);(2,4)(21,23,27,36)(22,28,31,43)(24,35,26,33)(25,34,39,50)(29,44,40,32)(30,42,46,49)(37,51,41,48)(38,52,45,47);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,24,27,26)(22,29,31,40)(23,33,36,35)(25,37,39,41)(28,32,43,44)(30,38,46,45)(34,48,50,51)(42,47,49,52);(7,15)(10,13)(14,20)(18,19)(21,25,27,39)(22,30,31,46)(23,34,36,50)(24,37,26,41)(28,42,43,49)(29,38,40,45)(32,47,44,52)(33,48,35,51)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6579357);