#418773

order	512
nilpotency class	3
isoclinism	isoclinic class of #418772
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15266
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1878 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,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	#72317787, #72318546, #72318589
maximal normal subgroup of	#72317787, #72318546, #72318589

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325969);