#472700

order	512
nilpotency class	3
isoclinism	isoclinic class of #457608
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2735
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16832
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

708 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)(3,14,18,39)(4,10)(5,25)(6,12)(7,13)(8,30)(9,17,33,19)(11,28)(15,43,20,41)(16,32,49,60)(21,37)(22,35)(23,36)(24,54)(26,55)(27,38)(29,40)(31,46,57,45)(34,52)(42,48,59,50)(44,63,51,56)(47,62,64,58)(53,61);(1,3,6,19,7,18,8,17)(2,9,12,39,13,33,30,14)(4,20,22,45,23,15,24,46)(5,16,28,48,29,49,27,50)(10,31,35,43,36,57,54,41)(11,32,40,59,38,60,25,42)(21,51,52,64,53,44,26,47)(34,56,61,58,55,63,37,62);(1,4,6,22,7,23,8,24)(2,10,12,35,13,36,30,54)(3,15,19,46,18,20,17,45)(5,26,27,53,29,52,28,21)(9,31,39,43,33,57,14,41)(11,37,25,55,38,61,40,34)(16,47,50,44,49,64,48,51)(32,58,42,56,60,62,59,63);(1,5)(2,11)(3,16)(4,21)(6,28)(7,29)(8,27)(9,32)(10,34)(12,40)(13,38)(14,42)(15,44)(17,50)(18,49)(19,48)(20,51)(22,52)(23,53)(24,26)(25,30)(31,56)(33,60)(35,61)(36,55)(37,54)(39,59)(41,62)(43,58)(45,64)(46,47)(57,63)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 379896);