#470605

order	512
nilpotency class	3
isoclinism	isoclinic class of #459756
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2761
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16690
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

874 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)(3,8,15,31)(4,17)(5,22,24,10)(6,25)(7,11)(9,20)(12,27)(13,38,18,48)(14,41)(16,46,45,30)(19,49,51,58)(21,37)(23,36,54,52)(26,34)(28,47,32,40)(29,43)(33,53,35,50)(39,59)(42,62)(44,56)(55,60)(57,63)(61,64);(1,3)(2,8,11,31)(4,18,21,13)(5,14)(6,16)(7,15)(9,32)(10,29,22,41)(12,30,25,46)(17,48)(19,42,51,39)(20,47,26,40)(23,44)(24,43)(27,45)(28,34)(33,57)(35,55)(36,56,52,61)(37,38)(49,62)(50,63,53,60)(54,64)(58,59);(1,4)(2,9)(3,13)(5,19)(6,26)(7,21)(8,28)(10,35)(11,34)(12,37)(14,42)(15,18)(16,47)(17,25)(20,27)(22,33)(23,53)(24,51)(29,55)(30,48)(31,32)(36,49)(38,46)(39,43)(40,45)(41,57)(44,60)(50,54)(52,58)(56,62)(59,61)(63,64);(1,5)(2,10)(3,14)(4,19)(6,23)(7,24)(8,29)(9,33)(11,22)(12,36)(13,39)(15,43)(16,44)(17,49)(18,42)(20,50)(21,51)(25,52)(26,53)(27,54)(28,55)(30,56)(31,41)(32,57)(34,35)(37,58)(38,59)(40,60)(45,64)(46,61)(47,63)(48,62)

Polycyclic presentation:

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

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, 377801);