#401791

order	512
nilpotency class	3
isoclinism	isoclinic class of #396106
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2459
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14823
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1135 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=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, 308987);