#149237

order	512
nilpotency class	4
isoclinism	isoclinic class of #13460
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2058
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11327
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

509 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=e,d^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,e^b=ei,g^b=gi,d^c=dh,f^c=fhi,h^c=hi

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, 56433);