#456890

order	512
nilpotency class	3
isoclinism	isoclinic class of #456890
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16287
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1109 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319592, #72319988, #72320118
maximal normal subgroup of	#72319592, #72319988, #72320118

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 364086);