#415238

order	512
nilpotency class	3
isoclinism	isoclinic class of #402003
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2494
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15187
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

633 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:

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

Polycyclic presentation:

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

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, 322434);