#149230

order	512
nilpotency class	4
isoclinism	isoclinic class of #135755
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1846
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11325
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

467 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)(3,14,19,44)(4,20,24,56)(5,26)(6,12)(7,31)(8,13)(9,16,35,53)(10,23,38,25)(11,29)(15,45)(17,42,18,43)(21,41)(22,39,27,37)(28,40)(30,60)(32,57)(33,51,34,52)(36,50)(46,55,58,59)(47,63)(48,62)(49,64)(54,61);(1,3,5,16,7,17,29,51,8,19,30,53,6,18,28,52)(2,9,11,14,12,34,26,43,13,35,40,44,31,33,60,42)(4,21,22,57,25,15,46,47,24,48,27,61,23,49,58,50)(10,36,37,45,20,54,55,41,38,63,39,64,56,32,59,62);(1,4)(2,10,13,38)(3,15,19,49)(5,27)(6,23)(7,25)(8,24)(9,32)(11,39,40,37)(12,20,31,56)(14,41)(16,50,53,47)(17,48,18,21)(22,30)(26,59,60,55)(28,46)(29,58)(33,63)(34,36)(35,54)(42,45)(43,64)(44,62)(51,57,52,61)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=eh,d^2=h,e^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,e^b=ei,g^b=gi,d^c=di,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, 56426);