#422927

order	512
nilpotency class	3
isoclinism	isoclinic class of #422927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15310
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1851 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)(3,15)(4,21)(5,27)(6,14)(7,12)(8,13)(9,18)(10,25)(11,30)(16,44)(17,48)(19,34)(20,35)(22,32)(23,53)(24,38)(26,37)(28,39)(29,41)(31,40)(33,50)(36,56)(42,47)(43,49)(45,62)(46,59)(51,55)(52,60)(54,58)(57,61)(63,64);(1,3,6,18,8,19,7,20)(2,9,12,15,13,35,14,34)(4,22,24,47,26,16,25,46)(5,28,29,50,31,49,30,17)(10,32,37,59,38,44,21,42)(11,39,40,48,41,43,27,33)(23,54,55,45,56,64,57,52)(36,60,51,63,53,62,61,58);(1,4,6,25,8,26,7,24)(2,10,12,21,13,38,14,37)(3,16,18,47,19,22,20,46)(5,23,30,55,31,56,29,57)(9,32,15,42,35,44,34,59)(11,36,27,51,41,53,40,61)(17,45,49,64,50,52,28,54)(33,58,43,60,48,63,39,62);(1,5)(2,11)(3,17)(4,23)(6,30)(7,29)(8,31)(9,33)(10,36)(12,27)(13,41)(14,40)(15,43)(16,45)(18,49)(19,50)(20,28)(21,51)(22,52)(24,57)(25,55)(26,56)(32,58)(34,39)(35,48)(37,61)(38,53)(42,60)(44,63)(46,54)(47,64)(59,62)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264906, #72264941, #72265047
maximal normal subgroup of	#72264906, #72264941, #72265047
maximal quotient of	#72265227

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330123);