#148740

order	512
nilpotency class	4
isoclinism	isoclinic class of #136630
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1875
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11208
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

459 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=d,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,e^c=eh,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, 55936);