#149402

order	512
nilpotency class	4
isoclinism	isoclinic class of #140094
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1919
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11378
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

435 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,d^2=hi,e^2=hi,f^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=ghi,h^b=hi,d^c=dhi,e^c=eh,f^c=fh,g^c=ghi,h^c=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, 56598);