#503555

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

724 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:

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

Polycyclic presentation:

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

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, 410751);