#149145

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	#11315
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

413 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=h,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,e^c=ei,g^c=gi

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, 56341);