#148413

order	512
nilpotency class	4
isoclinism	isoclinic class of #13334
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2011
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11123
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

663 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,h^a=hi,c^b=cf,e^b=ei,d^c=dh,f^c=fhi,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, 55609);