#148364

order	512
nilpotency class	4
isoclinism	isoclinic class of #13469
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2065
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11121
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

665 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=i,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,h^a=hi,c^b=cf,f^b=fi,g^b=gi,d^c=dh,e^c=ei,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, 55560);