#500886

order	512
nilpotency class	4
isoclinism	isoclinic class of #499908
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
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

732 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408082);