#149228

order	512
nilpotency class	4
isoclinism	isoclinic class of #135851
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	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

471 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)(3,14,18,41)(4,20,24,53)(5,26)(6,12)(7,31)(8,13)(9,16,34,48)(10,23,36,25)(11,29)(15,42)(17,40,19,50)(21,54)(22,37,27,35)(28,38)(30,58)(32,60)(33,47,59,49)(39,45)(43,51)(44,61)(46,63)(52,55,57,56)(62,64);(1,3,5,16,7,19,29,49,8,18,30,48,6,17,28,47)(2,9,11,14,12,33,26,40,13,34,38,41,31,59,58,50)(4,21,22,46,25,44,56,64,24,15,27,43,23,45,55,60)(10,32,35,61,20,51,52,54,36,62,37,39,53,63,57,42);(1,4)(2,10,13,36)(3,15)(5,27)(6,23)(7,25)(8,24)(9,32,34,62)(11,37,38,35)(12,20,31,53)(14,39,41,61)(16,46)(17,44)(18,21)(19,45)(22,30)(26,57,58,52)(28,56)(29,55)(33,51,59,63)(40,42,50,54)(43,48)(47,60)(49,64)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=eh,d^2=h,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,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, 56424);