#454667

order	512
nilpotency class	3
isoclinism	isoclinic class of #454664
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1743 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(5,6,12,16,11,18,10,17)(7,19,22,13,24,38,23,37)(8,14,29,41,28,43,27,42)(9,30,34,15,33,45,32,46)(20,39,50,35,51,47,25,57)(21,52,54,61,56,36,55,48)(26,58,60,40,59,44,31,62)(49,66,53,64,68,63,67,65);(1,2)(3,4)(5,7)(6,13)(8,25)(9,21)(10,23)(11,24)(12,22)(14,39)(15,36)(16,38)(17,19)(18,37)(20,29)(26,53)(27,51)(28,50)(30,61)(31,49)(32,55)(33,56)(34,54)(35,41)(40,65)(42,57)(43,47)(44,66)(45,48)(46,52)(58,63)(59,67)(60,68)(62,64);(2,4)(5,8,11,28)(6,14,18,43)(7,20,24,51)(9,31,33,60)(10,29,12,27)(13,35,37,57)(15,44,46,58)(16,42,17,41)(19,47,38,39)(21,53,56,67)(22,25,23,50)(26,34,59,32)(30,62,45,40)(36,64,52,65)(48,66,61,63)(49,55,68,54);(5,9,11,33)(6,15,18,46)(7,21,24,56)(8,26,28,59)(10,34,12,32)(13,36,37,52)(14,40,43,62)(16,30,17,45)(19,48,38,61)(20,49,51,68)(22,55,23,54)(25,53,50,67)(27,60,29,31)(35,63,57,66)(39,65,47,64)(41,58,42,44)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361863);