#433543

order	512
nilpotency class	3
isoclinism	isoclinic class of #433542
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15484
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

739 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)(4,9,20,34)(5,22)(6,25)(7,11)(10,23)(12,28)(14,32)(15,42,43,54)(17,31,26,38)(18,29)(19,48,49,60)(21,52,51,35)(24,39)(27,37)(30,44,36,45)(33,53,59,50)(40,61)(41,56)(46,64)(47,58)(55,63)(57,62);(1,3)(2,8)(4,18,20,14)(5,15)(6,26)(7,16)(9,32,34,29)(10,36)(11,13)(12,38)(17,28)(19,46,49,40)(21,41,51,47)(22,54)(23,45)(24,43)(25,31)(27,44)(30,37)(33,55,59,57)(35,56,52,58)(39,42)(48,61,60,64)(50,62,53,63);(1,4)(2,9,11,34)(3,14)(5,19)(6,21)(7,20)(8,29,13,32)(10,33,37,59)(12,35,25,52)(15,40)(16,18)(17,41)(22,48,39,60)(23,50)(24,49)(26,47)(27,53)(28,51)(30,55,36,57)(31,56,38,58)(42,61,54,64)(43,46)(44,62)(45,63);(1,5,7,24)(2,10,11,37)(3,15,16,43)(4,19,20,49)(6,27,28,23)(8,30,13,36)(9,33,34,59)(12,39,25,22)(14,40,18,46)(17,45,26,44)(21,53,51,50)(29,55,32,57)(31,42,38,54)(35,60,52,48)(41,63,47,62)(56,61,58,64)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340739);