#452527

order	512
nilpotency class	3
isoclinism	isoclinic class of #452526
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16051
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

867 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=f,c^2=ef,d^2=efi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359723);