#148300

order	512
nilpotency class	4
isoclinism	isoclinic class of #135850
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1854
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11095
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

653 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)(3,15,19,44)(4,21,25,10)(5,26)(6,14)(7,13)(8,12)(9,17,33,53)(11,29)(16,47)(18,46,20,45)(22,57)(23,59,27,61)(24,38,31,42)(28,40)(30,41)(32,51)(34,54,35,52)(36,58)(37,60,39,48)(43,50)(49,63)(55,64)(56,62);(1,3,5,17,7,20,29,54,8,19,30,53,6,18,28,52)(2,9,11,15,14,35,26,46,12,33,40,44,13,34,41,45)(4,22,23,58,24,50,60,51,25,55,27,62,31,16,48,49)(10,36,37,64,38,63,61,43,21,56,39,57,42,32,59,47);(1,4)(2,10)(3,16)(5,27)(6,31)(7,24)(8,25)(9,32)(11,39)(12,21)(13,42)(14,38)(15,43)(17,51)(18,55)(19,50)(20,22)(23,30)(26,59)(28,60)(29,48)(33,63)(34,56)(35,36)(37,40)(41,61)(44,47)(45,64)(46,57)(49,53)(52,58)(54,62)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,g^b=gi,d^c=di,f^c=fi,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, 55496);