#148272

order	512
nilpotency class	4
isoclinism	isoclinic class of #148270
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

774 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=di,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, 55468);