#148417

order	512
nilpotency class	4
isoclinism	isoclinic class of #13523
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2092
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

669 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)(3,16)(4,22,27,45)(5,29)(6,13)(7,12)(8,33)(9,15)(10,18)(11,25,42,28)(14,31)(17,50,53,64)(19,36)(20,35)(21,49)(23,57,59,63)(24,46,61,40)(26,43,60,41)(30,44)(32,48)(34,58,62,51)(37,56)(38,54)(39,55,47,52);(1,3,7,20,8,19,31,54,9,21,30,56,6,32,5,18)(2,10,29,48,13,37,44,49,15,38,14,36,33,35,12,16)(4,23,26,52,28,58,61,53,27,59,60,55,25,51,24,17)(11,39,43,57,22,50,46,34,42,47,41,63,45,64,40,62);(1,4)(2,11)(3,17)(5,26)(6,28)(7,24)(8,25)(9,27)(10,34)(12,43)(13,45)(14,46)(15,42)(16,47)(18,23)(19,55)(20,51)(21,53)(22,33)(29,40)(30,61)(31,60)(32,52)(35,63)(36,64)(37,57)(38,62)(39,49)(41,44)(48,50)(54,59)(56,58)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,d^2=hi,e^2=i,f^2=h,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,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, 55613);