#471448

order	512
nilpotency class	3
isoclinism	isoclinic class of #455699
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2797
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16797
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1078 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)(3,13)(4,9)(5,23)(6,27)(7,12)(8,16)(10,25)(11,29)(14,42)(15,47)(17,32)(18,33)(19,53)(20,37)(21,38)(22,36)(24,57)(26,41)(28,40)(30,52)(31,50)(34,45)(35,55)(39,54)(43,51)(44,61)(46,59)(48,60)(49,56)(58,63)(62,64);(1,3)(2,8,12,32)(4,14,21,46)(5,15)(6,16)(7,18)(9,34)(10,31,40,47)(11,13,27,33)(17,29)(19,44,54,63)(20,45,22,52)(23,56,41,43)(24,48,55,64)(25,49)(26,50)(28,51)(30,38)(35,61)(36,42)(37,59)(39,62)(53,60)(57,58);(1,4)(2,9)(3,14)(5,24)(6,22)(7,21)(8,30)(10,39)(11,37)(12,38)(13,42)(15,48)(16,52)(17,45)(18,46)(19,25)(20,29)(23,35)(26,55)(27,36)(28,54)(31,60)(32,34)(33,59)(40,53)(41,57)(43,61)(44,49)(47,62)(50,64)(51,63)(56,58);(1,5)(2,10)(3,15)(4,19)(6,28)(7,26)(8,31)(9,35)(11,41)(12,40)(13,43)(14,44)(16,51)(17,49)(18,50)(20,55)(21,54)(22,24)(23,27)(25,29)(30,58)(32,47)(33,56)(34,61)(36,53)(37,39)(38,57)(42,60)(45,64)(46,63)(48,52)(59,62)

Polycyclic presentation:

2^9|b^2=egh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,e^d=ei,f^d=fi,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, 378644);