#148248

order	512
nilpotency class	4
isoclinism	isoclinic class of #148246
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11087
commutator	C₂²×D₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

776 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55444);