#417039

order	512
nilpotency class	3
isoclinism	isoclinic class of #402003
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2494
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15239
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

619 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)(3,13,17,8)(4,19,21,9)(5,23)(6,11)(7,26)(10,25)(12,14)(15,27)(16,42,43,50)(18,29,36,35)(20,47,48,51)(22,31,37,46)(24,52)(28,44,34,57)(30,49,45,59)(32,38)(33,41)(39,60)(40,56)(53,61)(54,58)(55,64)(62,63);(1,3,14,36,6,17,7,18)(2,8,12,29,11,13,26,35)(4,15,22,40,21,41,37,58)(5,16,38,57,24,43,25,44)(9,27,46,56,19,33,31,54)(10,28,23,50,32,34,52,42)(20,39,49,61,48,62,59,64)(30,53,47,63,45,55,51,60);(1,4)(2,9,11,19)(3,15)(5,20)(6,21)(7,22)(8,27,13,33)(10,30,32,45)(12,31,26,46)(14,37)(16,39)(17,41)(18,40)(23,51,52,47)(24,48)(25,49)(28,53,34,55)(29,54,35,56)(36,58)(38,59)(42,63,50,60)(43,62)(44,61)(57,64);(1,5,7,25,6,24,14,38)(2,10,12,23,11,32,26,52)(3,16,18,44,17,43,36,57)(4,20,22,49,21,48,37,59)(8,28,29,50,13,34,35,42)(9,30,31,51,19,45,46,47)(15,39,40,61,41,62,58,64)(27,53,54,60,33,55,56,63)

Polycyclic presentation:

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

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, 324235);