#388125

order	512
nilpotency class	3
isoclinism	isoclinic class of #388123
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14516
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

884 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295321);