#402880

order	512
nilpotency class	3
isoclinism	isoclinic class of #394482
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2435
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14860
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1120 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:

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

Polycyclic presentation:

2^9|a^2=eg,d^2=ef,f^2=i,g^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,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi

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, 310076);