#401706

order	512
nilpotency class	3
isoclinism	isoclinic class of #394756
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2436
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14822
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1075 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:

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

Polycyclic presentation:

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

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, 308902);