#432886

order	512
nilpotency class	3
isoclinism	isoclinic class of #420065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2686
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15468
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

964 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)(4,18,21,9)(5,23)(6,27)(7,11)(8,16)(10,26)(12,30)(14,31,19,34)(15,44)(17,33)(20,51,52,60)(22,36,28,39)(24,55)(25,40)(29,38)(32,48)(35,53,49,54)(37,45)(41,46)(42,61,50,64)(43,58,47,57)(56,63,59,62);(1,3)(2,8,11,13)(4,19)(5,24,25,15)(6,17)(7,16)(9,34,18,31)(10,37)(12,33,27,41)(14,21)(20,42,52,50)(22,47)(23,44)(26,48,29,45)(28,43)(30,46)(32,38)(35,56)(36,58,39,57)(40,55)(49,59)(51,64)(53,62,54,63)(60,61);(1,4)(2,9)(3,14,16,19)(5,20)(6,28)(7,21)(8,31,13,34)(10,35)(11,18)(12,39)(15,42,24,50)(17,47,46,43)(22,30)(23,51)(25,52)(26,54)(27,36)(29,53)(32,56,37,59)(33,58,41,57)(38,49)(40,60)(44,61,55,64)(45,63,48,62);(1,5)(2,10)(3,15)(4,20)(6,29)(7,25)(8,32)(9,35)(11,38)(12,40)(13,37)(14,42)(16,24)(17,48)(18,49)(19,50)(21,52)(22,54)(23,27)(26,30)(28,53)(31,56)(33,44)(34,59)(36,51)(39,60)(41,55)(43,63)(45,46)(47,62)(57,64)(58,61)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^c=gi,h^c=hi,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, 340082);