#148353

order	512
nilpotency class	4
isoclinism	isoclinic class of #13391
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2033
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11096
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

649 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)(3,14)(4,21,25,37)(5,26)(6,12)(7,31)(8,13)(9,17)(10,24,36,30)(11,15)(16,46,22,56)(18,42)(19,43)(20,53)(23,35,27,38)(28,39)(29,58)(32,59,62,64)(33,50)(34,51)(40,48,61,52)(41,44)(45,60,57,55)(47,63,49,54);(1,3,15,44,7,20,29,51,8,19,28,50,6,18,5,17)(2,9,26,42,12,33,39,43,13,34,58,53,31,41,11,14)(4,22,57,59,24,52,27,47,25,16,45,64,30,48,23,49)(10,32,60,46,37,63,38,40,36,62,55,56,21,54,35,61);(1,4)(2,10)(3,16)(5,27)(6,30)(7,24)(8,25)(9,32)(11,38)(12,37)(13,36)(14,40)(15,45)(17,49)(18,52)(19,22)(20,48)(21,31)(23,29)(26,55)(28,57)(33,63)(34,62)(35,39)(41,54)(42,56)(43,61)(44,59)(46,53)(47,51)(50,64)(58,60)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,f^b=fi,d^c=di,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, 55549);