#148351

order	512
nilpotency class	4
isoclinism	isoclinic class of #135851
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	#11096
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

657 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)(3,13)(4,19,22,9)(5,24)(6,28)(7,11)(8,16)(10,14)(12,26)(15,43,20,36)(17,38)(18,49)(21,54,25,57)(23,33,55,51)(27,56)(29,46,50,44)(30,48)(31,59)(32,52,34,41)(35,42)(37,40)(39,47)(45,60,53,64)(58,63,61,62);(1,3,14,40,6,18,27,48,7,17,42,59,26,47,5,16)(2,8,24,39,12,31,35,38,11,30,56,49,28,37,10,13)(4,20,52,62,23,53,25,44,22,15,41,63,55,45,21,46)(9,29,57,64,33,58,34,36,19,50,54,60,51,61,32,43);(1,4)(2,9)(3,15)(5,25)(6,23)(7,22)(8,29)(10,34)(11,19)(12,33)(13,36)(14,41)(16,46)(17,20)(18,45)(21,27)(24,54)(26,55)(28,51)(30,50)(31,58)(32,35)(37,61)(38,43)(39,60)(40,62)(42,52)(44,48)(47,53)(49,64)(56,57)(59,63)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cf,f^b=fi,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, 55547);