#118294

order	512
nilpotency class	3
isoclinism	isoclinic class of #118289
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

543 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,f^b=fhi,d^c=di,f^c=fh,g^c=gh

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, 25490);