#401725

order	512
nilpotency class	3
isoclinism	isoclinic class of #394813
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

1081 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:

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

Polycyclic presentation:

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

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, 308921);