#149226

order	512
nilpotency class	4
isoclinism	isoclinic class of #135850
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

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

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,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, 56422);