#423432

order	512
nilpotency class	3
isoclinism	isoclinic class of #423431
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15314
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

1274 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)(4,9,19,32)(5,21)(6,24)(7,11)(10,23)(12,26)(13,27,17,44)(14,38)(15,29)(16,42)(18,47,48,59)(20,51,50,33)(22,35)(25,34)(28,39)(30,41)(31,52,58,49)(36,60,45,64)(37,62,46,54)(40,56)(43,55)(53,63,57,61);(1,3)(2,8)(4,17,19,13)(5,14)(6,16)(7,15)(9,27,32,44)(10,28)(11,29)(12,30)(18,36,48,45)(20,46,50,37)(21,38)(22,40)(23,39)(24,42)(25,43)(26,41)(31,57,58,53)(33,54,51,62)(34,55)(35,56)(47,64,59,60)(49,61,52,63);(1,4)(2,9,11,32)(3,13)(5,18)(6,20)(7,19)(8,27,29,44)(10,31,34,58)(12,33,24,51)(14,36)(15,17)(16,37)(21,47,35,59)(22,48)(23,49)(25,52)(26,50)(28,53,55,57)(30,54,42,62)(38,60,56,64)(39,61)(40,45)(41,46)(43,63);(1,5,7,22)(2,10,11,34)(3,14,15,40)(4,18,19,48)(6,25,26,23)(8,28,29,55)(9,31,32,58)(12,35,24,21)(13,36,17,45)(16,43,41,39)(20,52,50,49)(27,53,44,57)(30,56,42,38)(33,59,51,47)(37,63,46,61)(54,64,62,60)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286090, #72286154, #72286267
maximal normal subgroup of	#72286090, #72286154, #72286267

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330628);