#404528

order	512
nilpotency class	3
isoclinism	isoclinic class of #395172
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

975 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)(3,8)(4,17,19,34)(5,21)(6,11)(7,12)(9,20,32,24)(10,23)(13,36,38,45)(14,40)(15,28)(16,29)(18,47,48,46)(22,51)(25,35)(26,39,30,43)(27,42)(31,52,50,49)(33,53)(37,59,60,58)(41,62)(44,56)(54,63,57,61)(55,64);(1,3,6,15,7,16,25,44)(2,8,11,28,12,29,35,56)(4,13,24,43,19,38,20,39)(5,14,22,41,23,42,53,64)(9,30,34,36,32,26,17,45)(10,27,33,55,21,40,51,62)(18,37,52,63,48,60,49,61)(31,57,47,58,50,54,46,59);(1,4)(2,9)(3,13)(5,18)(6,24)(7,19)(8,26)(10,31)(11,34)(12,32)(14,37)(15,43)(16,38)(17,35)(20,25)(21,50)(22,52)(23,48)(27,54)(28,45)(29,30)(33,47)(36,56)(39,44)(40,57)(41,63)(42,60)(46,51)(49,53)(55,59)(58,62)(61,64);(1,5)(2,10,12,21)(3,14)(4,18)(6,22)(7,23)(8,27,29,40)(9,31,32,50)(11,33,35,51)(13,37)(15,41)(16,42)(17,46,34,47)(19,48)(20,49)(24,52)(25,53)(26,54,30,57)(28,55,56,62)(36,58,45,59)(38,60)(39,61)(43,63)(44,64)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285582, #72285784, #72285960, …
maximal normal subgroup of	#72285582, #72285784, #72285960, …
maximal quotient of	C₂×#404528

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311724);