#148435

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:

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

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=fhi,h^a=hi,c^b=cf,e^b=ei,f^b=fi,d^c=dhi,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, 55631);