#405080

order	512
nilpotency class	3
isoclinism	isoclinic class of #397903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2856
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14910
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

853 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=f,d^2=g,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,e^c=ei,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, 312276);