#404009

order	512
nilpotency class	3
isoclinism	isoclinic class of #404008
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14885
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23412	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2486 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)(3,15)(4,21)(5,25)(6,14)(7,12)(8,13)(9,18)(10,23)(11,26)(16,34)(17,31)(19,33)(20,32)(22,51)(24,39)(27,38)(28,37)(29,36)(30,45)(35,54)(40,44)(41,46)(42,60)(43,49)(47,57)(48,56)(50,53)(52,59)(55,61)(58,62)(63,64);(1,3)(2,9)(4,16)(5,17)(6,19)(7,18)(8,20)(10,34)(11,31)(12,32)(13,33)(14,15)(21,49)(22,42)(23,44)(24,45)(25,41)(26,48)(27,46)(28,47)(29,43)(30,36)(35,58)(37,56)(38,57)(39,40)(50,64)(51,55)(52,62)(53,61)(54,63)(59,60);(1,4)(2,10)(3,16)(5,22)(6,29)(7,23)(8,24)(9,30)(11,35)(12,39)(13,36)(14,21)(15,40)(17,42)(18,44)(19,43)(20,45)(25,50)(26,53)(27,52)(28,54)(31,55)(32,49)(33,34)(37,59)(38,51)(41,60)(46,62)(47,63)(48,61)(56,64)(57,58);(1,5,8,28)(2,11)(3,17,20,47)(4,22,24,54)(6,27,7,26)(9,31)(10,35)(12,37)(13,38)(14,25)(15,41)(16,42,45,63)(18,48,19,46)(21,50)(23,53,29,52)(30,55)(32,56)(33,57)(34,58)(36,51)(39,59)(40,60)(43,62,44,61)(49,64)

Polycyclic presentation:

2^9|d^2=gi,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,e^b=ei,h^b=hi,e^c=ei,f^c=fi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311205);