#417743

order	512
nilpotency class	3
isoclinism	isoclinic class of #417742
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	#60387	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2306 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)(3,15,19,42)(4,20,25,37)(5,26,30,40)(6,13,7,12)(9,31,34,18)(10,23,38,24)(11,28,41,29)(16,32,47,60)(17,33,49,57)(21,43,46,35)(22,52,56,36)(27,44,48,39)(45,63,53,61)(50,55,58,54)(51,62,64,59);(1,3)(2,9)(4,21)(5,27)(6,31)(7,18)(8,19)(10,35)(11,39)(12,42)(13,15)(14,34)(16,24)(17,28)(20,32)(22,53)(23,47)(25,46)(26,57)(29,49)(30,48)(33,40)(36,62)(37,60)(38,43)(41,44)(45,56)(50,61)(51,54)(52,59)(55,64)(58,63);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,25)(9,32)(11,36)(12,37)(13,20)(14,38)(15,43)(17,45)(18,46)(19,47)(21,31)(26,58)(27,51)(28,54)(29,55)(30,56)(33,59)(34,60)(35,42)(39,61)(40,50)(41,52)(44,63)(48,64)(49,53)(57,62);(1,5)(2,11)(3,17)(4,22)(6,29)(7,28)(8,30)(9,33)(10,36)(12,26)(13,40)(14,41)(15,44)(16,45)(18,48)(19,49)(20,50)(21,51)(23,55)(24,54)(25,56)(27,31)(32,59)(34,57)(35,61)(37,58)(38,52)(39,42)(43,63)(46,64)(47,53)(60,62)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318900, #72318925, #72318981
maximal normal subgroup of	#72318900, #72318925, #72318981

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324939);