#512437

order	512
nilpotency class	4
isoclinism	isoclinic class of #511960
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

514 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2,4,6)(3,7,8,5)(9,10,13,17)(11,23,26,48)(12,30,31,47)(14,35,34,46)(15,22,16,21)(18,41,38,27)(19,37,39,33)(20,36,40,24)(25,54,55,71)(28,51,29,50)(32,63,62,45)(42,67,65,72)(43,58,44,52)(49,59,70,57)(53,61,64,69)(56,60,66,68);(1,3,4,8)(2,5,6,7)(10,17)(18,38)(19,39)(20,40)(21,22)(23,48)(30,47)(35,46)(42,65)(43,44)(45,63)(49,70)(50,51)(54,71)(60,68)(61,69);(9,11,24,52,16,29,34,41,13,26,36,58,15,28,14,27)(10,18,35,51,22,44,40,48,17,38,46,50,21,43,20,23)(12,25,53,72,33,57,62,66,31,55,64,67,37,59,32,56)(19,42,61,54,30,60,63,70,39,65,69,71,47,68,45,49);(9,12)(10,19)(11,25)(13,31)(14,32)(15,37)(16,33)(17,39)(18,42)(20,45)(21,47)(22,30)(23,49)(24,53)(26,55)(27,56)(28,59)(29,57)(34,62)(35,61)(36,64)(38,65)(40,63)(41,66)(43,68)(44,60)(46,69)(48,70)(50,71)(51,54)(52,72)(58,67)

Polycyclic presentation:

2^9|a^2=e,b^2=ei,c^2=fh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,h^a=hi,g^d=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419633);