#147990

order	512
nilpotency class	3
isoclinism	isoclinic class of #2081
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#397
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11042
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

665 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,e^b=ei,d^c=di,f^c=fi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55186);