#424739

order	512
nilpotency class	3
isoclinism	isoclinic class of #417669
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2678
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15324
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1042 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)(4,21)(5,27,31,41)(6,13)(7,12)(8,14)(9,18)(10,25)(11,30,40,29)(16,32)(17,47,48,33)(19,35)(20,34)(22,51)(23,53,56,36)(24,38)(26,37)(28,39,49,43)(42,46)(44,63,52,61)(45,60)(50,57,62,55)(54,59,58,64);(1,3)(2,9)(4,22)(5,28)(6,19)(7,18)(8,20)(10,32)(11,39)(12,34)(13,15)(14,35)(16,26)(17,29)(21,42)(23,54)(24,45)(25,46)(27,33)(30,48)(31,49)(36,61)(37,60)(38,51)(40,43)(41,47)(44,57)(50,59)(52,55)(53,63)(56,58)(62,64);(1,4,7,25,8,26,6,24)(2,10,13,21,14,38,12,37)(3,16,19,46,20,22,18,45)(5,23,29,55,31,56,30,57)(9,32,34,60,35,51,15,42)(11,36,27,50,40,53,41,62)(17,44,28,58,48,52,49,54)(33,59,39,61,47,64,43,63);(1,5)(2,11)(3,17)(4,23)(6,30)(7,29)(8,31)(9,33)(10,36)(12,41)(13,27)(14,40)(15,43)(16,44)(18,49)(19,28)(20,48)(21,50)(22,52)(24,57)(25,55)(26,56)(32,59)(34,39)(35,47)(37,62)(38,53)(42,63)(45,54)(46,58)(51,64)(60,61)

Polycyclic presentation:

2^9|a^2=ef,c^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=gi,e^d=ei,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, 331935);