#500265

order	512
nilpotency class	4
isoclinism	isoclinic class of #500262
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24058
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#61666	C₂
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

825 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407461);