#148251

order	512
nilpotency class	4
isoclinism	isoclinic class of #148249
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

784 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=ei,f^a=fhi,c^b=cf,f^b=fhi,h^b=hi,d^c=dh,f^c=fi,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, 55447);