#148952

order	512
nilpotency class	4
isoclinism	isoclinic class of #13512
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2085
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11256
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

443 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)(4,18,21,9)(5,11)(6,12)(7,10)(8,15)(14,30,45,43)(16,44)(17,41)(19,33,52,50)(20,51,55,34)(22,42,29,35)(23,57)(24,59,56,39)(25,37)(26,40)(27,36)(28,32)(31,47)(38,58)(46,60,49,63)(48,62,53,64)(54,61);(1,3)(2,8,10,13)(4,19)(5,23)(6,28)(7,15)(9,33,18,50)(11,38,36,57)(12,41,37,32)(14,20)(16,26)(17,25)(21,52)(22,48)(24,46)(27,58)(29,53)(30,51,43,34)(31,61,44,40)(35,62,42,64)(39,60,59,63)(45,55)(47,54)(49,56);(1,4)(2,9)(3,14,15,45)(5,24)(6,29)(7,21)(8,30,13,43)(10,18)(11,39)(12,42)(16,48,47,53)(17,49,28,46)(19,23,52,58)(20,54)(22,25)(26,55)(27,56)(31,62,44,64)(32,63,41,60)(33,38,50,57)(34,61)(35,37)(36,59)(40,51)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,f^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=ei,c^b=cf,d^b=di,f^b=fh,h^b=hi,f^c=fhi,g^c=gi,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, 56148);