#402883

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

1126 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)(4,21,26,10)(5,27)(6,14)(7,12)(8,13)(9,18)(11,28)(16,35,43,48)(17,32)(19,33)(20,34)(22,31,47,58)(23,51,53,56)(24,37,25,38)(29,39)(30,40)(36,52,49,54)(41,46)(42,57,61,64)(44,55)(45,59)(50,63,62,60);(1,3)(2,9)(4,22)(5,17)(6,20)(7,18)(8,19)(10,35)(11,32)(12,33)(13,34)(14,15)(16,24)(21,48)(23,50)(25,43)(26,47)(27,55)(28,46)(29,44)(30,45)(31,37)(36,60)(38,58)(39,59)(40,41)(42,52)(49,63)(51,64)(53,62)(54,61)(56,57);(1,4)(2,10)(3,16)(5,23)(6,24)(7,25)(8,26)(9,31)(11,36)(12,37)(13,21)(14,38)(15,35)(17,42)(18,22)(19,43)(20,47)(27,56)(28,52)(29,54)(30,53)(32,57)(33,48)(34,58)(39,49)(40,51)(41,60)(44,50)(45,61)(46,62)(55,63)(59,64);(1,5,6,28,8,30,7,29)(2,11,14,40,13,39,12,27)(3,17,20,46,19,45,18,44)(4,23,24,52,26,53,25,54)(9,32,15,41,34,59,33,55)(10,36,38,51,21,49,37,56)(16,42,47,62,43,61,22,50)(31,57,35,60,58,64,48,63)

Polycyclic presentation:

2^9|a^2=eg,d^2=ef,e^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,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,g^c=gi,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, 310079);