#401056

order	512
nilpotency class	3
isoclinism	isoclinic class of #394108
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14811
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1649 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285493, C₂×#401056, #72285753, …
maximal normal subgroup of	#72285493, C₂×#401056, #72285753, …
maximal quotient of	C₂×#401056

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308252);