#454044

order	512
nilpotency class	3
isoclinism	isoclinic class of #454041
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3020	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2121 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)(7,18)(8,24)(9,15)(10,34)(11,16)(12,17)(13,21)(14,27)(19,35)(20,52)(22,37)(23,44)(25,38)(26,57)(28,40)(29,45)(30,60)(31,46)(32,42)(33,43)(36,54)(39,58)(41,59)(47,50)(48,56)(49,66)(51,62)(53,63)(55,64)(61,68)(65,67);(5,7)(6,13)(8,25)(9,20)(10,22)(11,21)(12,23)(14,38)(15,36)(16,44)(17,37)(18,34)(19,28)(24,35)(26,53)(27,51)(29,50)(30,49)(31,55)(32,54)(33,56)(39,65)(40,47)(41,66)(42,48)(43,64)(45,62)(46,52)(57,61)(58,68)(59,67)(60,63);(2,4)(5,8,12,29)(6,14,17,40)(7,19,23,51)(9,30,33,58)(10,28,11,27)(13,35,37,62)(15,41,43,57)(16,45,34,24)(18,47,44,38)(20,53,56,67)(21,25,22,50)(26,32,59,31)(36,63,64,65)(39,46,60,42)(48,66,52,61)(49,55,68,54);(5,9,12,33)(6,15,17,43)(7,20,23,56)(8,26,29,59)(10,32,11,31)(13,36,37,64)(14,39,40,60)(16,46,34,42)(18,48,44,52)(19,49,51,68)(21,55,22,54)(24,57,45,41)(25,53,50,67)(27,58,28,30)(35,61,62,66)(38,65,47,63)

Polycyclic presentation:

2^9|c^2=i,d^2=i,e^2=i,f^2=i,g^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,f^c=fi,g^c=gi,e^d=ei,f^d=fi,g^d=gi

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, 361240);