#148328

order	512
nilpotency class	4
isoclinism	isoclinic class of #134702
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1809
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

651 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)(4,20,23,37)(5,24)(6,12)(7,29)(8,13)(9,17)(10,22,36,28)(11,15)(16,45,48,63)(18,40)(19,41)(21,35,53,62)(25,38)(26,55)(27,42)(30,58,34,60)(31,43)(32,50)(33,56)(39,47,61,49)(44,52,57,54)(46,51,64,59);(1,3,15,43,7,18,26,50,8,19,25,56,6,27,5,17)(2,9,24,42,12,33,38,41,13,32,55,40,29,31,11,14)(4,16,44,60,22,49,53,64,23,48,57,58,28,47,21,46)(10,34,54,63,37,59,62,39,36,30,52,45,20,51,35,61);(1,4)(2,10)(3,16)(5,21)(6,28)(7,22)(8,23)(9,30)(11,35)(12,37)(13,36)(14,39)(15,44)(17,46)(18,49)(19,48)(20,29)(24,54)(25,57)(26,53)(27,47)(31,59)(32,34)(33,51)(38,62)(40,63)(41,61)(42,45)(43,60)(50,64)(52,55)(56,58)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,e^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,e^c=ei,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, 55524);