#404537

order	512
nilpotency class	3
isoclinism	isoclinic class of #395190
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2431
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14893
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

963 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,d^2=g,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,h^c=hi,e^d=ei,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, 311733);