#396106

order	512
nilpotency class	3
isoclinism	isoclinic class of #396106
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2459
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14730
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1364 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)(3,13,16,8)(4,9)(5,10,22,35)(6,24)(7,11)(12,27)(14,28,42,39)(15,40)(17,30,25,37)(18,46,45,31)(19,48,49,32)(20,38)(21,33)(23,52,51,36)(26,34)(29,44)(41,61)(43,56)(47,57)(50,58,54,60)(53,59)(55,62)(63,64);(1,3)(2,8)(4,18)(5,15)(6,25)(7,16)(9,31)(10,29)(11,13)(12,37)(14,20)(17,27)(19,47)(21,45)(22,44)(23,53)(24,30)(26,42)(28,34)(32,57)(33,46)(35,40)(36,59)(38,39)(41,50)(43,51)(48,64)(49,63)(52,56)(54,62)(55,58)(60,61);(1,4)(2,9)(3,14)(5,19)(6,26)(7,21)(8,28)(10,32)(11,33)(12,38)(13,39)(15,41)(16,42)(17,45)(18,25)(20,27)(22,49)(23,54)(24,34)(29,55)(30,46)(31,37)(35,48)(36,60)(40,61)(43,63)(44,62)(47,53)(50,51)(52,58)(56,64)(57,59);(1,5)(2,10,11,35)(3,15)(4,19)(6,23)(7,22)(8,29,13,40)(9,32,33,48)(12,36,24,52)(14,41)(16,44)(17,43)(18,47)(20,50)(21,49)(25,53)(26,54)(27,51)(28,55,39,61)(30,56,37,59)(31,57,46,64)(34,58,38,60)(42,62)(45,63)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 303302);