#497148

order	512
nilpotency class	4
isoclinism	isoclinic class of #497148
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#22276
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33722	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1747 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

552 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=efh,d^2=eh,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=dh,e^b=ei,g^b=gi,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404344);