#503299

order	512
nilpotency class	4
isoclinism	isoclinic class of #501072
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

698 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)(12,27,30,59)(14,34,33,65)(15,21,16,38)(18,43,45,31)(20,35,48,29)(23,51,53,69)(25,57,56,66)(26,42,36,41)(28,62,55,58)(32,60,37,61)(39,54,49,72)(40,52,50,67)(44,63,70,71)(46,68,47,64);(1,3,5,4)(2,6,8,7)(9,11,13,24)(10,17,19,22)(12,28,30,55)(14,25,33,56)(15,36,16,26)(18,44,45,70)(20,40,48,50)(21,42,38,41)(23,37,53,32)(27,58,59,62)(29,52,35,67)(31,63,43,71)(34,66,65,57)(39,47,49,46)(51,60,69,61)(54,68,72,64);(9,12,29,64,16,32,33,43,13,30,35,68,15,37,14,31)(10,18,34,61,21,47,48,59,19,45,65,60,38,46,20,27)(11,23,52,71,26,55,56,72,24,53,67,63,36,28,25,54)(17,39,66,58,42,44,50,51,22,49,57,62,41,70,40,69);(1,4,5,3)(2,7,8,6)(10,19)(17,22)(18,45)(20,48)(21,38)(27,59)(34,65)(39,49)(40,50)(41,42)(44,70)(46,47)(51,69)(57,66)(58,62)(60,61)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=fh,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,g^c=gi

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, 410495);