#480369

order	512
nilpotency class	4
isoclinism	isoclinic class of #480315
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×M₁₆	C₂⁴
socle	C₂	C₂×#1466
prime radical	#3449	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #2873 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

873 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=gh,f^2=h,h^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,d^b=di,f^b=fg,d^c=di,f^c=fi,f^e=fi

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, 387565);