#502765

order	512
nilpotency class	4
isoclinism	isoclinic class of #501262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28496	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

804 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=ei,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,g^c=ghi,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, 409961);