#147327

order	512
nilpotency class	4
isoclinism	isoclinic class of #13096
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1898
commutator	C₂²×C₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂	#10912
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

673 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:

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

Polycyclic presentation:

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

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, 54523);