#457252

order	512
nilpotency class	3
isoclinism	isoclinic class of #457252
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16291
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

1296 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)(3,15,18,9,19,34,31,35)(4,20,23,39,25,38,24,10)(5,26,28,41,30,11,29,42)(16,45,21,36,47,32,52,43)(17,48,50,56,51,44,59,33)(22,53,27,57,55,40,54,37)(46,62,49,61,58,60,64,63);(1,3,8,19)(2,9)(4,21,25,52)(5,17,30,51)(6,18,7,31)(10,36)(11,33)(12,15)(13,34)(14,35)(16,23,47,24)(20,32)(22,49,55,64)(26,56)(27,58,54,46)(28,50,29,59)(37,61)(38,45)(39,43)(40,62)(41,44)(42,48)(53,60)(57,63);(1,4)(2,10)(3,16)(5,27)(6,24)(7,23)(8,25)(9,32)(11,40)(12,20)(13,38)(14,39)(15,43)(17,49)(18,52)(19,47)(21,31)(22,28)(26,53)(29,55)(30,54)(33,61)(34,36)(35,45)(37,41)(42,57)(44,60)(46,50)(48,62)(51,64)(56,63)(58,59);(1,5,6,28,8,30,7,29)(2,11,12,41,14,26,13,42)(3,17,18,50,19,51,31,59)(4,22,23,27,25,55,24,54)(9,33,15,44,35,56,34,48)(10,37,38,40,39,57,20,53)(16,46,21,49,47,58,52,64)(32,60,36,61,45,62,43,63)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287261, #72287302, #72287334
maximal normal subgroup of	#72287261, #72287302, #72287334

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 364448);