#149035

order	512
nilpotency class	4
isoclinism	isoclinic class of #13432
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2044
commutator	C₂×#41	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂	#11287
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

602 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,h^a=hi,c^b=cf,d^b=dhi,f^b=fhi,g^b=ghi,h^b=hi,e^c=eh,f^c=fh,g^c=gh,h^c=hi,e^d=ei,f^d=fi,g^d=gi,f^e=fi,g^e=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, 56231);