#454098

order	512
nilpotency class	3
isoclinism	isoclinic class of #454095
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60368	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2136 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:

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

Polycyclic presentation:

2^9|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,h^a=hi,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, 361294);