#417621

order	512
nilpotency class	3
isoclinism	isoclinic class of #417620
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15257
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60398	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2330 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:

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

Polycyclic presentation:

2^9|a^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,f^b=fi,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	#72319954, #72319969, #72320088
maximal normal subgroup of	#72319954, #72319969, #72320088

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324817);