#430485

order	512
nilpotency class	3
isoclinism	isoclinic class of #417686
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15420
prime radical	#23414	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1149 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)(3,15,19,33)(4,10)(5,24,27,39)(6,12)(7,14)(8,13)(9,18,32,20)(11,28,40,26)(16,44,47,42)(17,48)(21,52,55,57)(22,36)(23,37)(25,43)(29,41)(30,51,34,46)(31,50)(35,54,61,56)(38,49)(45,59)(53,60)(58,63)(62,64);(1,3,6,20,8,19,7,18)(2,9,12,33,13,32,14,15)(4,16,22,51,23,47,29,46)(5,25,28,50,27,49,26,17)(10,34,36,42,37,30,41,44)(11,38,24,48,40,43,39,31)(21,53,56,63,55,64,54,45)(35,60,57,58,61,62,52,59);(1,4)(2,10)(3,16)(5,21)(6,22)(7,29)(8,23)(9,30)(11,35)(12,36)(13,37)(14,41)(15,42)(17,45)(18,46)(19,47)(20,51)(24,57)(25,53)(26,54)(27,55)(28,56)(31,58)(32,34)(33,44)(38,62)(39,52)(40,61)(43,60)(48,59)(49,64)(50,63);(1,5,6,28,8,27,7,26)(2,11,12,24,13,40,14,39)(3,17,20,25,19,50,18,49)(4,21,22,56,23,55,29,54)(9,31,33,38,32,48,15,43)(10,35,36,57,37,61,41,52)(16,45,51,53,47,63,46,64)(30,58,44,62,34,59,42,60)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 337681);