#470932

order	512
nilpotency class	3
isoclinism	isoclinic class of #458069
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2653
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16718
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

677 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=efh,c^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,h^b=hi,d^c=de,e^c=ei,e^d=ei,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, 378128);